Quiz about Analytic Trigonometry

Question 1

Complete the identity: $$\sec\ x\ -\frac{1}{\sec\ x\ }=?$$

Accepted Answer

sin x tan x

Answer

sec x csc x

Answer

1 + cot x

Answer

-2 tan x

Question 2

Complete the identity: $$\frac{\sin\ x}{\cos\ x}+\frac{\cos\ x}{\sin\ x}\ =\ ?$$

Accepted Answer

sec x csc x

Answer

sec x tan x

Answer

1 + cot x

Answer

sin x tan x

Question 3

Complete the identity: $$\sin^2x\ +\ \sin^2x\ \cot^2x\ =\ ?$$

Accepted Answer

1

Answer

$$\sin^2x\ +\ 1$$

Answer

$$\cot^2x+1$$

Answer

$$\cot^2x\ -\ 1$$

Question 4

Find the exact value of the expression: $$\cos\ 165\degree\ \cos\ 45\degree\ +\ \sin\ 165\degree\ \sin\ 45\degree\ =\ ?$$

Accepted Answer

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

Answer

$$-2$$

Answer

$$-\sqrt[]{3}$$

Answer

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

Question 5

Find the exact value of the expression $$\sin\ 10\degree\ \cos50\degree\ +\ \cos\ 10\degree\ \sin\ 50\degree.$$

Accepted Answer

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

Answer

$$\frac{\sqrt[]{3}}{3}$$

Answer

$$\frac{1}{6}$$

Answer

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

Question 6

Find the exact value of the expression $$\cos\ \frac{5\pi}{12}\ \sin\ \frac{\pi}{4}\ -\ \cos\ \frac{\pi}{4}\ \sin\ \frac{5\pi}{4}.$$

Accepted Answer

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

Answer

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

Answer

$$\frac{1}{4}$$

Answer

$$1$$

Question 7

Use the given information to find the exact value of the expression:

$$\sin\alpha\ =\frac{5}{13},\ \alpha\ lies\ in\ Quadrant\ 1,\ and\ \cos\ \beta\ =\frac{15}{17}\ lies\ in\ Quadrant\ 1.\ Find\ \ \cos\ \left(\alpha\ +\beta\right).$$

Accepted Answer

$$\frac{140}{221}$$

Answer

$$-\frac{21}{221}$$

Answer

$$\frac{220}{221}$$

Answer

$$\frac{171}{221}$$

Question 8

Given $$ an\ \alpha=\frac{21}{20},\ \alpha\ lies\ in\ Quadrant\ III,\ and\ \cos\ \beta\ =-\frac{5}{13},\ \beta\ lies\ in\ Quadrant\ II.\ Find\ \sin\ \left(\alpha\ +\ \beta ight).$$

Accepted Answer

$$-\frac{135}{377}$$

Answer

$$\frac{352}{377}$$

Answer

$$-\frac{152}{377}$$

Answer

$$\frac{345}{377}$$

Question 9

Find the exact value of tan $$\frac{\pi}{12}$$ using a difference formula.

Accepted Answer

$$2-\sqrt[]{3}$$

Answer

$$2+\sqrt[]{3}$$

Answer

$$\sqrt[]{3}-2$$

Answer

$$-2-\sqrt[]{3}$$

Question 10

Find the exact value: $$\sin\ \alpha=\frac{4}{5},\ 0<\alpha<\frac{\pi}{2};\ \cos\ \beta=\frac{8}{17},\ 0<\beta<\frac{\pi}{2}.\ Find\ an\ \left(\alpha+\beta ight).$$

Accepted Answer

$$-\frac{77}{36}$$

Answer

$$-\frac{36}{85}$$

Answer

$$\frac{84}{85}$$

Answer

$$\frac{77}{85}$$

Question 11

Find $$\sin\ 2 heta.$$

Accepted Answer

$$\frac{840}{841}$$

Answer

$$-\frac{41}{841}$$

Answer

$$\frac{39}{841}$$

Answer

$$\frac{41}{841}$$

Question 12

Express the product as a sum or difference: sin 7x cos 5x.

Accepted Answer

$$\frac{1}{2}\left(\sin\ 12x\ +\ \sin\ 2x\right)$$

Answer

$$\frac{1}{2}\left(\cos\ 12x\ -\ \cos\ 2x\right)$$

Answer

$$\frac{1}{2}\left(\sin\ 12x\ +\ \cos\ 2x\right)$$

Answer

$$\sin\ \left(\cos\ 35x^2\right)$$

Question 13

Express the expression $$\sin\ 9x\ -\ \sin\ 3x$$ as a product.

Accepted Answer

$$2\ \sin\ 3x\ \cos\ 6x$$

Answer

$$2\ \cos\ 3x\ \cos\ 6x$$

Answer

$$2\ \sin\ 6x\ \cos\ 3x$$

Answer

$$2\ \sin\ 3x$$

Question 14

Solve the equation $$\cos^2x\ +\ 2\cos\ x\ +\ 1\ =\ 0$$ on the interval $$\left[0,\ 2\pi\right).$$

Accepted Answer

$$\pi$$

Answer

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

Answer

$$\frac{\pi}{2},\ \frac{3\pi}{2}$$

Answer

$$2\pi$$

Question 15

Solve the equation $$2\ \sin\ heta\ +\ 3\ =\ 2$$ on the interval $$\left[0,\ 2\pi ight).$$

Accepted Answer

$$\left\{\frac{7\pi}{6},\ \frac{11\pi}{6}\right\}$$

Answer

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

Answer

$$\left\{\frac{\pi}{6},\ \frac{5\pi}{6},\ \frac{7\pi}{6},\ \frac{11\pi}{6}\right\}$$

Answer

$$\left\{\frac{4\pi}{3},\ \frac{2\pi}{3}\right\}$$

Analytic Trigonometry

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