Quiz about Trigonometric Identities

Question 1

Which expression is equal to the following?

$$\sin\left(- heta ight)\cos\left(- heta ight)$$

Accepted Answer

$$-\sin\left( heta ight)\cos\left( heta ight)$$

Answer

$$\sin\left( heta ight)\cos\left( heta ight)$$

Answer

$$ an\left( heta ight)$$

Answer

$$- an\left( heta ight)$$

Question 2

Which expression is equal to the following?

$$2\sin\left(-x ight) an\left(-x ight)$$

Accepted Answer

$$\frac{2\sin^2\left(x\right)}{\cos\left(x\right)}$$

Answer

$$-2\sin\left(x ight) an\left(x ight)$$

Answer

$$\frac{-2\sin^2\left(x\right)}{\cos\left(x\right)}$$

Answer

$$2\cos\left(x\right)$$

Question 3

Which expression is equal to the following?

$$\cos\left(x ight) an\left(x ight)+\sin\left(x ight)\cot\left(x ight)$$

Accepted Answer

$$\sin\left(x\right)+\cos\left(x\right)$$

Answer

$$\sin\left(x\right)-\cos\left(x\right)$$

Answer

$$1$$

Answer

$$ an\left(x ight)+\cot\left(x ight)$$

Question 4

Which of the following expressions is equal to $$\sin^2\left(x\right)$$ ?

Accepted Answer

$$\frac{1}{\csc^2\left(x\right)}$$

Answer

$$1+\cos^2\left(x\right)$$

Answer

$$\cos^2\left(x\right)-1$$

Answer

$$\frac{1}{\sec^2\left(x\right)}$$

Question 5

Which interval is correct for $$\frac{ heta}{2}$$ ?

$$\pi\le heta\le\frac{3\pi}{2}\ $$

Accepted Answer

$$\frac{\pi}{2}\le\frac{ heta}{2}\le\frac{3\pi}{4}\ $$

Answer

$$\frac{\pi}{2}\le\frac{ heta}{2}\le\frac{3\pi}{2}\ $$

Answer

$$\pi\le\frac{ heta}{2}\le\frac{3\pi}{2}\ $$

Answer

$$2\pi\le\frac{ heta}{2}\le3\pi\ $$

Question 6

Which expression is equal to the following?

$$\csc^2\left(x\right)-7\csc\left(x\right)+10$$

Accepted Answer

$$\left(\csc\left(x\right)-2\right)\left(\csc\left(x\right)-5\right)$$

Answer

$$\left(\csc\left(x\right)+2\right)\left(\csc\left(x\right)+5\right)$$

Answer

$$\left(\csc\left(x\right)-10\right)\left(\csc\left(x\right)+3\right)$$

Answer

$$\left(\sin\left(x\right)-2\right)\left(\sin\left(x\right)-5\right)$$

Question 7

Which expression is equal to the following from the double-angle identity?
$$\sin\left(180\right)$$

Accepted Answer

$$2\sin\left(90\right)\cos\left(90\right)$$

Answer

$$2\sin\left(360\right)\cos\left(360\right)$$

Answer

$$2\sin\left(180\right)\cos\left(180\right)$$

Answer

$$2\cos^2\left(90\right)-1$$

Question 8

Use a Pythagorean identity to find $$ an heta$$ .

$$\sec heta=2$$ and $$\sin heta<0$$

Accepted Answer

$$ an heta=-\sqrt{3}$$

Answer

$$ an heta=\frac{\sqrt{5}}{2}$$

Answer

$$ an heta=\frac{\sqrt{3}}{2}$$

Answer

$$ an heta=-\frac{\sqrt{3}}{2}$$

Question 9

Use a quotient identity to find $$\cot heta$$ .

$$\sin heta=\frac{4}{5}$$ and $$\sec heta=\frac{5}{3}$$

Accepted Answer

$$\cot heta=\frac{3}{4}$$

Answer

$$\cot heta=\frac{4}{3}$$

Answer

$$\cot heta=\frac{3}{5}$$

Answer

$$\cot heta=\frac{5}{4}$$

Question 10

Which expression is equal to the following?

$$\sin\left(\alpha-\frac{\pi}{2}\right)$$

Accepted Answer

$$-\cos\left(\alpha\right)$$

Answer

$$\cos\left(\alpha\right)$$

Answer

$$\sec\left(\alpha\right)$$

Answer

$$-\sec\left(\alpha\right)$$

Question 11

Which expression is equal to the following?

$$\frac{ an\left(10 ight)+ an\left(35 ight)}{1- an\left(10 ight) an\left(35 ight)}$$

Accepted Answer

$$ an\left(45 ight)$$

Answer

$$ an\left(25 ight)$$

Answer

$$ an\left(-25 ight)$$

Answer

$$ an\left(-45 ight)$$

Question 12

Evaluate the following using a sum identity.

$$\cos\left(10\right)\cos\left(80\right)-\sin\left(10\right)\sin\left(80\right)$$

Accepted Answer

0

Answer

1

Answer

90

Answer

-1

Question 13

Use a double-angle identity to find an expression for $$ an2 heta$$ .

$$\sin heta=\frac{1}{3}$$ and $$\cos heta=\frac{2}{3}$$

Accepted Answer

$$\frac{2\left(\frac{1}{2}\right)}{1-\left(\frac{1}{2}\right)^2}$$

Answer

$$\frac{2+\left(\frac{1}{2}\right)}{\left(\frac{1}{2}\right)^2}$$

Answer

$$\frac{2\left(\frac{1}{2}\right)}{1-\left(\frac{1}{2}\right)^{ }}$$

Answer

$$\frac{2\left(\frac{1}{2}\right)}{1+\left(\frac{1}{2}\right)^2}$$

Question 14

Use a double-angle identity to evaluate the expression.

$$2\cos^2\left(15\right)-1$$

Accepted Answer

$$\frac{\sqrt{3}}{2}$$

Answer

$$-\frac{\sqrt{3}}{2}$$

Answer

$$\frac{1}{2}$$

Answer

$$-\frac{1}{2}$$

Question 15

From the half-angle identities, which expression equals the following?

$$ an\left(22.5 ight)$$

Accepted Answer

$$\frac{1-\cos\left(45\right)}{\sin\left(45\right)}$$

Answer

$$\frac{\sin\left(22.5\right)}{1+\cos\left(22.5\right)}$$

Answer

$$\sqrt{\frac{1+\cos\left(45\right)}{1-\cos\left(45\right)}}$$

Answer

$$\frac{2 an\left(45 ight)}{1- an^2\left(45 ight)}$$

Question 16

From the half-angle identities, which expression is equal to the following?

$$\sin\left(67.5\right)$$

Accepted Answer

$$\sqrt{\frac{1-\cos\left(135\right)}{2}}$$

Answer

$$-\sqrt{\frac{1-\cos\left(135\right)}{2}}$$

Answer

$$\sqrt{\frac{1+\cos\left(135\right)}{2}}$$

Answer

$$2\sin\left(135\right)\cos\left(135\right)$$

Question 17

From the power reducing identities, which expression is equal to the following?

$$\cos^2\left(\frac{\pi}{8}\right)$$

Accepted Answer

$$\frac{1+\cos\left(\frac{\pi}{4}\right)}{2}$$

Answer

$$\frac{1+\cos\left(\frac{\pi}{16}\right)}{2}$$

Answer

$$\frac{1+\cos\left(\frac{\pi}{8}\right)}{2}$$

Answer

$$\frac{1-\cos\left(\frac{\pi}{4}\right)}{2}$$

Question 18

Which expression is equal to the following?

$$\sin\left(2x\right)+\sin\left(10x\right)$$

Accepted Answer

$$2\sin\left(6x\right)\cos\left(4x\right)$$

Answer

$$2\sin\left(4x\right)\cos\left(6x\right)$$

Answer

$$2\cos\left(6x\right)\cos\left(4x\right)$$

Answer

$$\sin\left(12x\right)$$

Question 19

Find all solutions to the following equation in the interval $$\left[\frac{\pi}{2},\frac{3\pi}{2}\right]$$ .

$$8\csc\left(\alpha\right)-11=5$$

Accepted Answer

$$\alpha=\frac{5\pi}{6}$$

Answer

$$\alpha=\frac{2\pi}{3}$$

Answer

$$\alpha\in\left\{\frac{\pi}{6},\frac{5\pi}{6}\right\}$$

Answer

$$\alpha\in\left\{\frac{\pi}{3},\frac{5\pi}{3}\right\}$$

Question 20

Find all solutions to the following equation in the interval $$\left[0,2\pi\right]$$

$$0=1-2\sin^2x$$

Accepted Answer

$$\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}$$

Answer

$$\frac{\pi}{4},\frac{3\pi}{4}$$

Answer

$$\frac{\pi}{4}$$

Answer

$$\frac{\pi}{4}+2\pi n,\frac{3\pi}{4}+2\pi n$$

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