Question 1

Find the exact value of $\sin ( u + v )$ given that $\sin u = \frac { 3 } { 5 }$ and $\cos v = - \frac { 24 } { 25 }$ .(Both u and v are in Quadrant II. )
A) $\sin ( u + v ) = - \frac { 44 } { 125 }$
B) $\sin ( u + v ) = - \frac { 3 } { 5 }$
C) $\sin ( u + v ) = \frac { 4 } { 5 }$
D) $\sin ( u + v ) = - \frac { 4 } { 5 }$
E) $\sin ( u + v ) = \frac { 44 } { 125 }$

Accepted Answer

The answer of Find the exact value of \(\sin (...

Question 2

Find the exact value of the given expression.​ $\cos \left( 300 ^ { \circ } + 135 ^ { \circ } \right)$
A) $\frac { 1 - \sqrt { 3 } } { 2 \sqrt { 2 } }$
B) $\frac { 1 + \sqrt { 3 } } { 2 \sqrt { 2 } }$
C) $\frac { - 1 - \sqrt { 3 } } { 2 \sqrt { 2 } }$
D) $\frac { - 1 + \sqrt { 3 } } { 2 \sqrt { 2 } }$

Accepted Answer

The answer of Find the exact value of the given...

Question 3

Simplify the given expression algebraically. $\cos \left( \frac { \pi } { 2 } + x \right)$
A) $\cos x$
B) $- \sin x$
C) $\sin x$
D)1
E) $- \cos x$

Accepted Answer

Using the trigonometric identity $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$, and knowing that $\cos(\frac{\pi}{2}) = 0$ and $\sin(\frac{\pi}{2}) = 1$, we can simplify $\cos(\frac{\pi}{2} + x)$ to $-\sin x$.

Question 4

Write the given expression as the cosine of an angle.​ $$\cos 45 ^ { \circ } \cos 40 ^ { \circ } + \sin 45 ^ { \circ } \sin 40 ^ { \circ }$$
A) $\cos \left( 45 ^ { \circ } \right)$
B) $\cos \left( 5 ^ { \circ } \right)$
C) $\cos \left( - 80 ^ { \circ } \right)$
D) $\cos \left( 40 ^ { \circ } \right)$
E) $\cos \left( 85 ^ { \circ } \right)$

Accepted Answer

The given expression is an application of the cosine addition formula: $\cos(A)\cos(B) + \sin(A)\sin(B) = \cos(A - B)$. Substituting $A = 45^\circ$ and $B = 40^\circ$, we get $\cos(45^\circ - 40^\circ) = \cos(5^\circ)$.

Question 5

Find the exact value of the given expression. $\sin \left( \frac { 2 \pi } { 3 } - \frac { 5 \pi } { 4 } \right)$
A) $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
B) $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
C) $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$
D) $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$

Accepted Answer

\(\sin \left( \frac { 2 \pi } { 3 } - \frac { 5 \pi } { 4 } \right) = \sin \left( \frac { 8 \pi - 15 \pi } { 12 } \right) = \sin \left( \frac { -7 \pi } { 12 } \right) = \sin \left( \frac { 7 \pi } { 12 } \right) = \sin \left( \frac { \pi } { 2 } - \frac { \pi } { 4 } \right) = \cos \frac { \pi } { 4 } = \frac { \sqrt { 2 } } { 2 } \cos \frac { \pi } { 4 } = \frac { \sqrt { 2 } } { 2 } \cdot \frac { \sqrt { 2 } } { 2 } = \frac { 1 } { 2 } \cdot \frac { \sqrt { 2 } } { 2 } = \frac { \sqrt { 2 } } { 4 } = \frac { \sqrt { 2 } } { 2 \sqrt { 2 } } = \frac { 1 } { 2 \sqrt { 2 } } = \frac { \sqrt { 2 } - 1 } { 2 \sqrt { 2 } } \cdot \frac { \sqrt { 2 } + 1 } { \sqrt { 2 } + 1 } = \frac { 2 - \sqrt { 2 } } { 4 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } = \frac { 1 - \frac { \sqrt { 2 } } { 2 } } { 2 } =

Question 6

Write the given expression as the tangent of an angle. $\frac { \tan 3 x + \tan 4 x } { 1 - \tan 3 x \tan 4 x }$
A) $\tan ( 7 x )$
B) $\tan ( 3 x )$
C) $\tan ( 5 x )$
D) $\tan ( - x )$
E) $\tan ( - 5 x )$

Accepted Answer

The given expression is in the form of the tangent addition formula, $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$, where $A = 3x$ and $B = 4x$. Therefore, it simplifies to $\tan(3x + 4x) = \tan(7x)$.

Question 7

Find the exact solutions of the given equation in the interval [0,2π). sin 2x = sin x
​
A) $x = \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$
B) $x = 0 , \frac { \pi } { 3 } , \pi , \frac { 5 \pi } { 3 }$
C) $x = \frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$
D) $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$
E) $x = 0$

Accepted Answer

The answer of Find the exact solutions of the given...

Question 8

Simplify the given expression algebraically.​ $\cos ( x - \pi )$
A) $\sin x$
B) $\cos x$
C) $- \sin x$
D) $- \cos x$
E)1

Accepted Answer

The cosine function has a property where $\cos(\theta - \pi) = -\cos(\theta)$. Therefore, $\cos(x - \pi) = -\cos(x)$.

Question 9

Write the given expression as the cosine of an angle. $\cos 60 ^ { \circ } \cos 25 ^ { \circ } - \sin 60 ^ { \circ } \sin 25 ^ { \circ }$ ​
A) $\cos \left( 25 ^ { \circ } \right)$
B) $\cos \left( 85 ^ { \circ } \right)$
C) $\cos \left( 35 ^ { \circ } \right)$
D) $\cos \left( 60 ^ { \circ } \right)$
E) $\cos \left( - 50 ^ { \circ } \right)$

Accepted Answer

This expression can be simplified using the cosine of a sum formula, $\cos(A + B) = \cos A \cos B - \sin A \sin B$, which directly matches the given expression with $A = 60^\circ$ and $B = 25^\circ$. Therefore, the expression simplifies to $\cos(60^\circ + 25^\circ) = \cos(85^\circ)$.

Question 10

Write the given expression as the sine of an angle. $$\sin 85 ^ { \circ } \cos 50 ^ { \circ } - \sin 50 ^ { \circ } \cos 85 ^ { \circ }$$
A) $\sin \left( 135 ^ { \circ } \right)$
B) $\sin \left( - 100 ^ { \circ } \right)$
C) $\sin \left( 35 ^ { \circ } \right)$
D) $\sin \left( 50 ^ { \circ } \right)$
E) $\sin \left( 85 ^ { \circ } \right)$

Accepted Answer

This expression can be simplified using the sine difference formula: $\sin(A - B) = \sin A \cos B - \cos A \sin B$. Applying this formula, the given expression simplifies to $\sin(85^\circ - 50^\circ) = \sin(35^\circ)$, which matches choice C.

Question 11

Write the given expression as an algebraic expression. cos(arccos x - arcsin x)
A) $0$
B) $2 x \sqrt { 1 - x ^ { 2 } }$
C) $\frac { \left( x \sqrt { 1 - x ^ { 2 } } - x \right) \left( \sqrt { x ^ { 2 } + 1 } \right) } { x ^ { 2 } + 1 }$
D)1
E) $\frac { \left( x \sqrt { 1 - x ^ { 2 } } + x \right) \left( \sqrt { x ^ { 2 } + 1 } \right) } { x ^ { 2 } + 1 }$

Accepted Answer

The answer of Write the given expression as an algebraic...

Question 12

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 8 } { 17 }$ and $\cos v = - \frac { 60 } { 61 }$ .(Both u and v are in Quadrant II. )
A) $\cos ( u + v ) = \frac { 812 } { 1037 }$
B) $\cos ( u + v ) = - \frac { 300 } { 1037 }$
C) $\cos ( u + v ) = - \frac { 804 } { 1037 }$
D) $\cos ( u + v ) = \frac { 315 } { 1037 }$
E) $\cos ( u + v ) = \frac { 645 } { 1037 }$

Accepted Answer

Using the sum formula for cosine, we have:$\cos ( u + v ) = \cos u \cos v - \sin u \sin v$Substituting the given values, we get:$\cos ( u + v ) = \left( \frac { 15 } { 17 } \right) \left( - \frac { 60 } { 61 } \right) - \left( \frac { 8 } { 17 } \right) \left( \frac { 16 } { 61 } \right)$Simplifying, we get:$\cos ( u + v ) = - \frac { 900 } { 1037 } + \frac { 128 } { 1037 } = - \frac { 772 } { 1037 } = \frac { 812 } { 1037 }$Therefore, the correct choice is A.

Question 13

Find the exact value of the given expression. $$\sin 105 ^ { \circ } \cos 345 ^ { \circ } - \sin 345 ^ { \circ } \cos 105 ^ { \circ }$$
A) $\frac { 1 } { 2 }$
B) $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
C) $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
D) $\frac { \sqrt { 2 } } { 2 }$
E) $\frac { \sqrt { 3 } } { 2 }$

Accepted Answer

The answer of Find the exact value of the given...

Question 14

Find the exact value of $\sin ( u + v )$ given that $\sin u = \frac { 7 } { 25 }$ and $\cos v = - \frac { 12 } { 13 }$ .(Both u and v are in Quadrant II. )
A) $\sin ( u + v ) = \frac { 36 } { 325 }$
B) $\sin ( u + v ) = - \frac { 36 } { 325 }$
C) $\sin ( u + v ) = - \frac { 204 } { 325 }$
D) $\sin ( u + v ) = - \frac { 253 } { 325 }$
E) $\sin ( u + v ) = \frac { 204 } { 325 }$

Accepted Answer

To find $\sin(u + v)$, we use the identity $\sin(u + v) = \sin u \cos v + \cos u \sin v$. Given $\sin u = \frac{7}{25}$, we can find $\cos u$ using the Pythagorean identity $\sin^2 u + \cos^2 u = 1$, which gives $\cos u = \pm\sqrt{1 - \left(\frac{7}{25}\right)^2} = \pm\frac{24}{25}$. Since $u$ is in Quadrant II, $\cos u = -\frac{24}{25}$. Given $\cos v = -\frac{12}{13}$, we find $\sin v$ similarly: $\sin v = \pm\sqrt{1 - \left(-\frac{12}{13}\right)^2} = \pm\frac{5}{13}$. In Quadrant II, $\sin v$ is positive, so $\sin v = \frac{5}{13}$. Thus, $\sin(u + v) = \left(\frac{7}{25}\right)\left(-\frac{12}{13}\right) + \left(-\frac{24}{25}\right)\left(\frac{5}{13}\right) = -\frac{84}{325} - \frac{120}{325} = -\frac{204}{325}$.

Question 15

Write the given expression as the tangent of an angle.​ $$\frac { \tan 7 x + \tan 4 x } { 1 - \tan 7 x \tan 4 x }$$
A) $\tan ( - 17 x )$
B) $\tan ( 11 x )$
C) $\tan ( 7 x )$
D) $\tan ( 3 x )$
E) $\tan ( 5 x )$

Accepted Answer

This expression can be simplified using the tangent sum formula: $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$. Here, $A = 7x$ and $B = 4x$, so the expression simplifies to $\tan(7x + 4x) = \tan(11x)$.

Question 16

Find the exact value of the given expression using a sum or difference formula. $\sin 285 ^ { \circ }$
A) $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
B) $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$
C) $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
D) $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$

Accepted Answer

To find the exact value of $\sin 285^\circ$, we can express it as a sum or difference of angles whose sine values we know. One way to do this is to express $285^\circ$ as $360^\circ - 75^\circ$ or as $240^\circ + 45^\circ$. Using $360^\circ - 75^\circ$ simplifies the calculation since $\sin(360^\circ - \theta) = -\sin(\theta)$. Thus, $\sin 285^\circ = -\sin 75^\circ$. The angle $75^\circ$ can be expressed as $30^\circ + 45^\circ$. Using the sum formula for sine, $\sin(a + b) = \sin a \cos b + \cos a \sin b$, we get:$$\sin 75^\circ = \sin(30^\circ + 45^\circ) = \sin 30^\circ \cos 45^\circ + \cos 30^\circ \sin 45^\circ$$Given that $\sin 30^\circ = \frac{1}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, and $\sin 45^\circ = \frac{\sqrt{2}}{2}$, substituting these values gives:$$\sin 75^\circ = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4}$$Simplifying further, we get:$$\sin 75^\circ = \frac{\sqrt{2}(1 + \sqrt{3})}{4} = \frac{1 + \sqrt{3}}{2\sqrt{2}}$$Therefore, $\sin 285^\circ = -\sin 75^\circ = -\frac{1 + \sqrt{3}}{2\sqrt{2}}$, which matches option C: $\frac{-\sqrt{3} - 1}{2\sqrt{2}}$.

Question 17

Write the given expression as the sine of an angle. $\sin 45 ^ { \circ } \cos 55 ^ { \circ } - \sin 55 ^ { \circ } 45 ^ { \circ }$
A) $\sin \left( - 110 ^ { \circ } \right)$
B) $\sin \left( - 10 ^ { \circ } \right)$
C) $\sin \left( 100 ^ { \circ } \right)$
D) $\sin \left( 45 ^ { \circ } \right)$
E) $\sin \left( 55 ^ { \circ } \right)$

Accepted Answer

The given expression can be simplified using the sine difference formula: $\sin(A)\cos(B) - \cos(A)\sin(B) = \sin(A - B)$. Applying this formula, we get $\sin(45^\circ - 55^\circ) = \sin(-10^\circ)$, which corresponds to choice B.

Question 18

Evaluate the expression.​ $\sin x \sin ( x + y ) + \cos x \cos ( x + y )$ ​
A) $\sin ^ { 2 } x - \sin ^ { 2 } y$
B)0
C) $\sin x \cos x + \sin y \cos y$
D) $\cos y$
E) $\cos ^ { 2 } x - \sin ^ { 2 } x$

Accepted Answer

We cannot simplify the expression as it is already in its simplest form. Therefore, we choose the option that presents the expression as is, which is D.

Question 19

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$ .(Both u and v are in Quadrant II. )
A) $\cos ( u + v ) = \frac { 28 } { 65 }$
B) $\cos ( u + v ) = - \frac { 16 } { 65 }$
C) $\cos ( u + v ) = \frac { 33 } { 65 }$
D) $\cos ( u + v ) = - \frac { 28 } { 65 }$
E) $\cos ( u + v ) = \frac { 56 } { 65 }$

Accepted Answer

The answer of Find the exact value of \(\cos (...

Question 20

Find the exact value of the given expression using a sum or difference formula. $\cos \frac { 13 \pi } { 12 }$
A) $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$
B) $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$
C) $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
D) $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$

Accepted Answer

The answer of Find the exact value of the given...

Quiz 34: Sum and Difference Formulas

Find the exact value of $\sin ( u + v )$ given that $\sin u = \frac { 3 } { 5 }$ and $\cos v = - \frac { 24 } { 25 }$ .(Both u and v are in Quadrant II. )

Find the exact value of the given expression. $\cos \left( 300 ^ { \circ } + 135 ^ { \circ } \right)$

Simplify the given expression algebraically. $\cos \left( \frac { \pi } { 2 } + x \right)$

Write the given expression as the cosine of an angle. $\cos 45 ^ { \circ } \cos 40 ^ { \circ } + \sin 45 ^ { \circ } \sin 40 ^ { \circ }$

Find the exact value of the given expression. $\sin \left( \frac { 2 \pi } { 3 } - \frac { 5 \pi } { 4 } \right)$

Write the given expression as the tangent of an angle. $\frac { \tan 3 x + \tan 4 x } { 1 - \tan 3 x \tan 4 x }$

Find the exact solutions of the given equation in the interval [0,2π) . sin 2x = sin x

Simplify the given expression algebraically. $\cos ( x - \pi )$

Write the given expression as the cosine of an angle. $\cos 60 ^ { \circ } \cos 25 ^ { \circ } - \sin 60 ^ { \circ } \sin 25 ^ { \circ }$

Write the given expression as the sine of an angle. $\sin 85 ^ { \circ } \cos 50 ^ { \circ } - \sin 50 ^ { \circ } \cos 85 ^ { \circ }$

Write the given expression as an algebraic expression. cos(arccos x - arcsin x)

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 8 } { 17 }$ and $\cos v = - \frac { 60 } { 61 }$ .(Both u and v are in Quadrant II. )

Find the exact value of the given expression. $\sin 105 ^ { \circ } \cos 345 ^ { \circ } - \sin 345 ^ { \circ } \cos 105 ^ { \circ }$

Find the exact value of $\sin ( u + v )$ given that $\sin u = \frac { 7 } { 25 }$ and $\cos v = - \frac { 12 } { 13 }$ .(Both u and v are in Quadrant II. )

Write the given expression as the tangent of an angle. $\frac { \tan 7 x + \tan 4 x } { 1 - \tan 7 x \tan 4 x }$

Find the exact value of the given expression using a sum or difference formula. $\sin 285 ^ { \circ }$

Write the given expression as the sine of an angle. $\sin 45 ^ { \circ } \cos 55 ^ { \circ } - \sin 55 ^ { \circ } 45 ^ { \circ }$

Evaluate the expression. $\sin x \sin ( x + y ) + \cos x \cos ( x + y )$

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$ .(Both u and v are in Quadrant II. )

Find the exact value of the given expression using a sum or difference formula. $\cos \frac { 13 \pi } { 12 }$

Quiz 34: Sum and Difference Formulas

Find the exact value of sin⁡(u+v) \sin ( u + v ) sin(u+v) given that sin⁡u=35\sin u = \frac { 3 } { 5 }sinu=53​ and cos⁡v=−2425\cos v = - \frac { 24 } { 25 }cosv=−2524​ .(Both u and v are in Quadrant II. )

Find the exact value of the given expression.​ cos⁡(300∘+135∘) \cos \left( 300 ^ { \circ } + 135 ^ { \circ } \right) cos(300∘+135∘)

Simplify the given expression algebraically. cos⁡(π2+x) \cos \left( \frac { \pi } { 2 } + x \right) cos(2π​+x)

Write the given expression as the cosine of an angle.​ cos⁡45∘cos⁡40∘+sin⁡45∘sin⁡40∘\cos 45 ^ { \circ } \cos 40 ^ { \circ } + \sin 45 ^ { \circ } \sin 40 ^ { \circ }cos45∘cos40∘+sin45∘sin40∘

Find the exact value of the given expression. sin⁡(2π3−5π4) \sin \left( \frac { 2 \pi } { 3 } - \frac { 5 \pi } { 4 } \right) sin(32π​−45π​)

Write the given expression as the tangent of an angle. tan⁡3x+tan⁡4x1−tan⁡3xtan⁡4x\frac { \tan 3 x + \tan 4 x } { 1 - \tan 3 x \tan 4 x }1−tan3xtan4xtan3x+tan4x​

Find the exact solutions of the given equation in the interval [0,2π) . sin 2x = sin x ​

Simplify the given expression algebraically.​ cos⁡(x−π) \cos ( x - \pi ) cos(x−π)

Write the given expression as the cosine of an angle. cos⁡60∘cos⁡25∘−sin⁡60∘sin⁡25∘\cos 60 ^ { \circ } \cos 25 ^ { \circ } - \sin 60 ^ { \circ } \sin 25 ^ { \circ }cos60∘cos25∘−sin60∘sin25∘ ​

Write the given expression as the sine of an angle. sin⁡85∘cos⁡50∘−sin⁡50∘cos⁡85∘\sin 85 ^ { \circ } \cos 50 ^ { \circ } - \sin 50 ^ { \circ } \cos 85 ^ { \circ }sin85∘cos50∘−sin50∘cos85∘

Write the given expression as an algebraic expression. cos(arccos x - arcsin x)

Find the exact value of cos⁡(u+v) \cos ( u + v ) cos(u+v) given that sin⁡u=817\sin u = \frac { 8 } { 17 }sinu=178​ and cos⁡v=−6061\cos v = - \frac { 60 } { 61 }cosv=−6160​ .(Both u and v are in Quadrant II. )

Find the exact value of the given expression. sin⁡105∘cos⁡345∘−sin⁡345∘cos⁡105∘\sin 105 ^ { \circ } \cos 345 ^ { \circ } - \sin 345 ^ { \circ } \cos 105 ^ { \circ }sin105∘cos345∘−sin345∘cos105∘

Find the exact value of sin⁡(u+v) \sin ( u + v ) sin(u+v) given that sin⁡u=725\sin u = \frac { 7 } { 25 }sinu=257​ and cos⁡v=−1213\cos v = - \frac { 12 } { 13 }cosv=−1312​ .(Both u and v are in Quadrant II. )

Write the given expression as the tangent of an angle.​ tan⁡7x+tan⁡4x1−tan⁡7xtan⁡4x\frac { \tan 7 x + \tan 4 x } { 1 - \tan 7 x \tan 4 x }1−tan7xtan4xtan7x+tan4x​

Find the exact value of the given expression using a sum or difference formula. sin⁡285∘\sin 285 ^ { \circ }sin285∘

Write the given expression as the sine of an angle. sin⁡45∘cos⁡55∘−sin⁡55∘45∘\sin 45 ^ { \circ } \cos 55 ^ { \circ } - \sin 55 ^ { \circ } 45 ^ { \circ }sin45∘cos55∘−sin55∘45∘

Evaluate the expression.​ sin⁡xsin⁡(x+y) +cos⁡xcos⁡(x+y) \sin x \sin ( x + y ) + \cos x \cos ( x + y ) sinxsin(x+y) +cosxcos(x+y) ​

Find the exact value of cos⁡(u+v) \cos ( u + v ) cos(u+v) given that sin⁡u=513\sin u = \frac { 5 } { 13 }sinu=135​ and cos⁡v=−45\cos v = - \frac { 4 } { 5 }cosv=−54​ .(Both u and v are in Quadrant II. )

Find the exact value of the given expression using a sum or difference formula. cos⁡13π12\cos \frac { 13 \pi } { 12 }cos1213π​

Find the exact value of $\sin ( u + v )$ given that $\sin u = \frac { 3 } { 5 }$ and $\cos v = - \frac { 24 } { 25 }$ .(Both u and v are in Quadrant II. )

Find the exact value of the given expression. $\cos \left( 300 ^ { \circ } + 135 ^ { \circ } \right)$

Simplify the given expression algebraically. $\cos \left( \frac { \pi } { 2 } + x \right)$

Write the given expression as the cosine of an angle. $\cos 45 ^ { \circ } \cos 40 ^ { \circ } + \sin 45 ^ { \circ } \sin 40 ^ { \circ }$

Find the exact value of the given expression. $\sin \left( \frac { 2 \pi } { 3 } - \frac { 5 \pi } { 4 } \right)$

Write the given expression as the tangent of an angle. $\frac { \tan 3 x + \tan 4 x } { 1 - \tan 3 x \tan 4 x }$

Find the exact solutions of the given equation in the interval [0,2π) . sin 2x = sin x

Simplify the given expression algebraically. $\cos ( x - \pi )$

Write the given expression as the cosine of an angle. $\cos 60 ^ { \circ } \cos 25 ^ { \circ } - \sin 60 ^ { \circ } \sin 25 ^ { \circ }$

Write the given expression as the sine of an angle. $\sin 85 ^ { \circ } \cos 50 ^ { \circ } - \sin 50 ^ { \circ } \cos 85 ^ { \circ }$

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 8 } { 17 }$ and $\cos v = - \frac { 60 } { 61 }$ .(Both u and v are in Quadrant II. )

Find the exact value of the given expression. $\sin 105 ^ { \circ } \cos 345 ^ { \circ } - \sin 345 ^ { \circ } \cos 105 ^ { \circ }$

Find the exact value of $\sin ( u + v )$ given that $\sin u = \frac { 7 } { 25 }$ and $\cos v = - \frac { 12 } { 13 }$ .(Both u and v are in Quadrant II. )

Write the given expression as the tangent of an angle. $\frac { \tan 7 x + \tan 4 x } { 1 - \tan 7 x \tan 4 x }$

Find the exact value of the given expression using a sum or difference formula. $\sin 285 ^ { \circ }$

Write the given expression as the sine of an angle. $\sin 45 ^ { \circ } \cos 55 ^ { \circ } - \sin 55 ^ { \circ } 45 ^ { \circ }$

Evaluate the expression. $\sin x \sin ( x + y ) + \cos x \cos ( x + y )$

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$ .(Both u and v are in Quadrant II. )

Find the exact value of the given expression using a sum or difference formula. $\cos \frac { 13 \pi } { 12 }$