Question 1

Find all solutions of the equation $2 \sin ^{2} x+3 \sin x-2=0$ on the interval $[0,2 \pi)$ (Your answer should be exact.)

Accepted Answer

$x=\frac{\pi}{6}, x=\frac{5 \pi}{6}$

Question 2

Prove the identity:
$\sin 3 x=4 \sin x \cos ^{2} x-\sin x$

Accepted Answer

$\begin{aligned}
\sin 3 x &=\sin x \cos 2 x+\cos x \sin 2 x \
&=\sin x\left(2 \cos ^{2} x-1\right)+\cos x(2 \sin x \cos x) \
&=4 \sin x \cos ^{2} x-\sin x
\end{aligned}$

Question 3

A golf ball is hit with initial velocity $v_{o}$ (in feet per second) and trajectory angle  $\theta .$ The horizontal distance traveled before it bounces is given by $d=\frac{v_{o}^{2}}{16} \sin \theta \cos \theta$
Find the angle of trajectory needed to hit a green 800 feet away on the first bounce if the initial velocity is  200 ft/sec

Accepted Answer

$\theta \approx 19.90^{\circ} \text { or } \theta \approx 70.10^{\circ}$

Question 4

The angles of elevation to the top of a tower from two points that are on the same side of the tower and 30 meters apart are $39.2^{\circ}$ and $42.6^{\circ}$ What is the height of the tower?

Accepted Answer

The answer of The angles of elevation to the top...

Question 5

The angle of elevation to the bottom of a steeple of a church is $35.2^{\circ}$ and the angle of elevation to the top of the steeple is $48.7^{\circ}$. If the angles of elevation are measured from a point on the ground 60 meters from the church steeple, what is the height of the steeple?

Accepted Answer

The answer of The angle of elevation to the bottom...

Question 6

Use the grapher to determine which of the following is an identity.

Accepted Answer

A)  $\sin ^{2} x-\cos x=1+2 \sin x$ 
B)  $\sin ^{3} x-\sin x=-\cos ^{2} x \sin x$ 
C)  $\sin (x+\pi)=\sin x$ 
D)  $\tan 2 x=2 \sin x \cos x$ 
E)  $\cos 2 x+\sin 2 x=1$ 
A)  $\sin ^{2} x-\cos x=1+2 \sin x$ 
B)  $\sin ^{3} x-\sin x=-\cos ^{2} x \sin x$ 
C)  $\sin (x+\pi)=\sin x$ 
D)  $\tan 2 x=2 \sin x \cos x$ 
E)  $\cos 2 x+\sin 2 x=1$

Question 7

Use the fundamental identities to change the expression to one involving only sines and cosines. Then simplify. Show all your steps.
$\sec ^{2} x-\frac{\sec ^{2} x}{\csc ^{2} x}$

Accepted Answer

The answer of Use the fundamental identities to change the...

Question 8

Prove the identity: 
$(1+\cot x)^{2}=\frac{1+2 \sin x \cos x}{\sin ^{2} x}$

Accepted Answer

None

Question 9

Simplify the expression $\frac{\csc \theta}{\sin ^{2} \theta+\sec \theta+\cos ^{2} \theta-1}$

Accepted Answer

The answer of Simplify the expression \(\frac{\csc \theta}{\sin ^{2} \theta+\sec...

Question 10

Prove the identity: 
$\cos 3 x=\cos ^{3} x-3 \sin ^{2} x \cos x$

Accepted Answer

None

Question 11

Use a power-reduction formula to find the exact value $\sin \left(\frac{\pi}{8}\right)$  (Hint: What is  $\sin ^{2}\left(\frac{\pi}{8}\right)$=?)

Accepted Answer

The answer of Use a power-reduction formula to find the...

Question 12

What is the area of   if  a=8, b=10 , and  c=15  ?

Accepted Answer

The answer of What is the area of  ...

Question 13

What is the general solution to $\cos ^{2} x \sin x-\sin x-\cos ^{2} x+1=0 ?$

Accepted Answer

The answer of What is the general solution to \(\cos...

Question 14

Use a sum or difference identity to find the exact value of  cos  $195^{\circ}$

Accepted Answer

The answer of Use a sum or difference identity to...

Question 15

Randy must find the distance between points  B  and  C  on opposite sides of a lake. He locates a point  A  that is  385ft from  B  and  546 ft from  C . If the angle at  A  is $81^{\circ}$ what is the distance  BC  ?

Accepted Answer

Approximat

Question 16

Use the sum identities to prove the identity: $\cos \theta+\cos 2 \theta+\cos 3 \theta=(2 \cos \theta+1) \cos 2 \theta$

Accepted Answer

None

Question 17

Prove the identity: 
$(\tan x+1)^{2}=\frac{1+2 \sin x \cos x}{\cos ^{2} x}$

Accepted Answer

The answer of Prove the identity: 
\((\tan x+1)^{2}=\frac{1+2 \sin x...

Question 18

A golf ball is hit with initial velocity $v_{o}$ (in feet per second) and trajectory angle $\theta$  The horizontal distance traveled before it bounces is given by $d=\frac{v_{o}^{2}}{16} \sin \theta \cos \theta$ Find the angle of trajectory needed to hit a green 700 feet away on the first bounce if the initial velocity is  170 ft/sec.

Accepted Answer

The answer of A golf ball is hit with initial...

Question 19

If  a=8, b=10 , and $\alpha=67^{\circ}$ how many triangles are determined?

Accepted Answer

The answer of If  a=8, b=10 , and $\alpha=67^{\circ}$...

Question 20

If  a=10, b=8 , and $\alpha=67^{\circ},$ how many triangles are determined?

Accepted Answer

The answer of If  a=10, b=8 , and $\alpha=67^{\circ},$...

Find all solutions of the equation $2 \sin ^{2} x+3 \sin x-2=0$ on the interval $[0,2 \pi)$ (Your answer should be exact.)

Prove the identity: $\sin 3 x=4 \sin x \cos ^{2} x-\sin x$

The angles of elevation to the top of a tower from two points that are on the same side of the tower and 30 meters apart are $39.2^{\circ}$ and $42.6^{\circ}$ What is the height of the tower?

Use the grapher to determine which of the following is an identity.

Use the fundamental identities to change the expression to one involving only sines and cosines. Then simplify. Show all your steps. $\sec ^{2} x-\frac{\sec ^{2} x}{\csc ^{2} x}$

Prove the identity: $(1+\cot x)^{2}=\frac{1+2 \sin x \cos x}{\sin ^{2} x}$

Simplify the expression $\frac{\csc \theta}{\sin ^{2} \theta+\sec \theta+\cos ^{2} \theta-1}$

Prove the identity: $\cos 3 x=\cos ^{3} x-3 \sin ^{2} x \cos x$

Use a power-reduction formula to find the exact value $\sin \left(\frac{\pi}{8}\right)$ (Hint: What is $\sin ^{2}\left(\frac{\pi}{8}\right)$ =?)

What is the area of if a=8, b=10 , and c=15 ?

What is the general solution to $\cos ^{2} x \sin x-\sin x-\cos ^{2} x+1=0 ?$

Use a sum or difference identity to find the exact value of cos $195^{\circ}$

Randy must find the distance between points B and C on opposite sides of a lake. He locates a point A that is 385ft from B and 546 ft from C . If the angle at A is $81^{\circ}$ what is the distance BC ?

Use the sum identities to prove the identity: $\cos \theta+\cos 2 \theta+\cos 3 \theta=(2 \cos \theta+1) \cos 2 \theta$

Prove the identity: $(\tan x+1)^{2}=\frac{1+2 \sin x \cos x}{\cos ^{2} x}$

If a=8, b=10 , and $\alpha=67^{\circ}$ how many triangles are determined?

If a=10, b=8 , and $\alpha=67^{\circ},$ how many triangles are determined?

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Discrete Mathematics

An Introduction to Calculus: Limits, Derivatives, and Integrals

P Prerequisites

Mathematical Problem Set

Filters

Exam 5: Analytic Trigonometry

Find all solutions of the equation 2sin⁡2x+3sin⁡x−2=02 \sin ^{2} x+3 \sin x-2=02sin2x+3sinx−2=0 on the interval [0,2π)[0,2 \pi)[0,2π) (Your answer should be exact.)

Prove the identity: sin⁡3x=4sin⁡xcos⁡2x−sin⁡x\sin 3 x=4 \sin x \cos ^{2} x-\sin xsin3x=4sinxcos2x−sinx

The angles of elevation to the top of a tower from two points that are on the same side of the tower and 30 meters apart are 39.2∘39.2^{\circ}39.2∘ and 42.6∘42.6^{\circ}42.6∘ What is the height of the tower?

Use the grapher to determine which of the following is an identity.

Use the fundamental identities to change the expression to one involving only sines and cosines. Then simplify. Show all your steps. sec⁡2x−sec⁡2xcsc⁡2x\sec ^{2} x-\frac{\sec ^{2} x}{\csc ^{2} x}sec2x−csc2xsec2x​

Prove the identity: (1+cot⁡x)2=1+2sin⁡xcos⁡xsin⁡2x(1+\cot x)^{2}=\frac{1+2 \sin x \cos x}{\sin ^{2} x}(1+cotx)2=sin2x1+2sinxcosx​

Simplify the expression csc⁡θsin⁡2θ+sec⁡θ+cos⁡2θ−1\frac{\csc \theta}{\sin ^{2} \theta+\sec \theta+\cos ^{2} \theta-1}sin2θ+secθ+cos2θ−1cscθ​

Prove the identity: cos⁡3x=cos⁡3x−3sin⁡2xcos⁡x\cos 3 x=\cos ^{3} x-3 \sin ^{2} x \cos xcos3x=cos3x−3sin2xcosx

Use a power-reduction formula to find the exact value sin⁡(π8)\sin \left(\frac{\pi}{8}\right)sin(8π​) (Hint: What is sin⁡2(π8)\sin ^{2}\left(\frac{\pi}{8}\right)sin2(8π​) =?)

What is the area of if a=8, b=10 , and c=15 ?

What is the general solution to cos⁡2xsin⁡x−sin⁡x−cos⁡2x+1=0?\cos ^{2} x \sin x-\sin x-\cos ^{2} x+1=0 ?cos2xsinx−sinx−cos2x+1=0?

Use a sum or difference identity to find the exact value of cos 195∘195^{\circ}195∘

Randy must find the distance between points B and C on opposite sides of a lake. He locates a point A that is 385ft from B and 546 ft from C . If the angle at A is 81∘81^{\circ}81∘ what is the distance BC ?

Use the sum identities to prove the identity: cos⁡θ+cos⁡2θ+cos⁡3θ=(2cos⁡θ+1)cos⁡2θ\cos \theta+\cos 2 \theta+\cos 3 \theta=(2 \cos \theta+1) \cos 2 \thetacosθ+cos2θ+cos3θ=(2cosθ+1)cos2θ

Prove the identity: (tan⁡x+1)2=1+2sin⁡xcos⁡xcos⁡2x(\tan x+1)^{2}=\frac{1+2 \sin x \cos x}{\cos ^{2} x}(tanx+1)2=cos2x1+2sinxcosx​

If a=8, b=10 , and α=67∘\alpha=67^{\circ}α=67∘ how many triangles are determined?

If a=10, b=8 , and α=67∘,\alpha=67^{\circ},α=67∘, how many triangles are determined?

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Discrete Mathematics

An Introduction to Calculus: Limits, Derivatives, and Integrals

P Prerequisites

Mathematical Problem Set

Filters

Find all solutions of the equation $2 \sin ^{2} x+3 \sin x-2=0$ on the interval $[0,2 \pi)$ (Your answer should be exact.)

Prove the identity: $\sin 3 x=4 \sin x \cos ^{2} x-\sin x$

The angles of elevation to the top of a tower from two points that are on the same side of the tower and 30 meters apart are $39.2^{\circ}$ and $42.6^{\circ}$ What is the height of the tower?

Use the fundamental identities to change the expression to one involving only sines and cosines. Then simplify. Show all your steps. $\sec ^{2} x-\frac{\sec ^{2} x}{\csc ^{2} x}$

Prove the identity: $(1+\cot x)^{2}=\frac{1+2 \sin x \cos x}{\sin ^{2} x}$

Simplify the expression $\frac{\csc \theta}{\sin ^{2} \theta+\sec \theta+\cos ^{2} \theta-1}$

Prove the identity: $\cos 3 x=\cos ^{3} x-3 \sin ^{2} x \cos x$

Use a power-reduction formula to find the exact value $\sin \left(\frac{\pi}{8}\right)$ (Hint: What is $\sin ^{2}\left(\frac{\pi}{8}\right)$ =?)

What is the general solution to $\cos ^{2} x \sin x-\sin x-\cos ^{2} x+1=0 ?$

Use a sum or difference identity to find the exact value of cos $195^{\circ}$

Randy must find the distance between points B and C on opposite sides of a lake. He locates a point A that is 385ft from B and 546 ft from C . If the angle at A is $81^{\circ}$ what is the distance BC ?

Use the sum identities to prove the identity: $\cos \theta+\cos 2 \theta+\cos 3 \theta=(2 \cos \theta+1) \cos 2 \theta$

Prove the identity: $(\tan x+1)^{2}=\frac{1+2 \sin x \cos x}{\cos ^{2} x}$

If a=8, b=10 , and $\alpha=67^{\circ}$ how many triangles are determined?

If a=10, b=8 , and $\alpha=67^{\circ},$ how many triangles are determined?