Trigonometric Identities: Master the Art of Trig Substitutions

Question 1

Expand and simplify the expression: (sin x + cos x)^2.

Accepted Answer

1 + 2sin x * cos x

Answer

1 + sin x + cos x

Answer

1 - sin x - cos x

Answer

1 - 2sin x * cos x

Question 2

Identify the trigonometric identity that states the relationship between sine and cosine of an angle:

Accepted Answer

Pythagorean identity: sin^2(x) + cos^2(x) = 1

Answer

Double-angle identity: cos(2x) = 2cos^2(x) - 1

Answer

Half-angle identity: tan(x/2) = (1 - cos(x)) / sin(x)

Answer

Sum-to-product identity: sin(a) + sin(b) = 2cos((a - b) / 2)sin((a + b) / 2)

Question 3

Which identity allows you to express the cosine of a double angle in terms of the original angle?

Accepted Answer

Double-angle identity: cos(2x) = 2cos^2(x) - 1

Answer

Pythagorean identity: sin^2(x) + cos^2(x) = 1

Answer

Half-angle identity: tan(x/2) = (1 - cos(x)) / sin(x)

Answer

Sum-to-product identity: sin(a) + sin(b) = 2cos((a - b) / 2)sin((a + b) / 2)

Question 4

Select the trigonometric identity that is useful when working with half angles:

Accepted Answer

Half-angle identity: tan(x/2) = (1 - cos(x)) / sin(x)

Answer

Pythagorean identity: sin^2(x) + cos^2(x) = 1

Answer

Double-angle identity: cos(2x) = 2cos^2(x) - 1

Answer

Sum-to-product identity: sin(a) + sin(b) = 2cos((a - b) / 2)sin((a + b) / 2)

Question 5

Which identity is specifically designed for finding the sine of a double angle?

Accepted Answer

Double-angle identity: sin(2x) = 2sin(x)cos(x)

Answer

Pythagorean identity: sin^2(x) + cos^2(x) = 1

Answer

Half-angle identity: tan(x/2) = (1 - cos(x)) / sin(x)

Answer

Sum-to-product identity: sin(a) + sin(b) = 2cos((a - b) / 2)sin((a + b) / 2)

Question 6

If the tangent of an angle x is equal to 2, what is the value of its secant?

Accepted Answer

√5

Answer

1 / √5

Answer

2

Answer

1 / 2

Question 7

Which of the following identities represents the Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

tan²θ + 1 = sec²θ

Answer

cot²θ - 1 = csc²θ

Answer

sin²θ = tan²θ + cot²θ

Question 8

Identify the correct double-angle identity for cosine.

Accepted Answer

cos(2θ) = 2cos²θ - 1

Answer

cos(2θ) = cos²θ - sin²θ

Answer

cos(2θ) = 1 - 2sin²θ

Answer

cos(2θ) = sin²θ + cos²θ

Question 9

Which identity is used to find the half-angle of sine?

Accepted Answer

sin(θ/2) = √((1 - cosθ) / 2)

Answer

cos(θ/2) = √((1 + cosθ) / 2)

Answer

tan(θ/2) = √((1 - cosθ) / (1 + cosθ))

Answer

cot(θ/2) = √((1 + cosθ) / (1 - cosθ))

Question 10

Determine the value of cos(π/6) using trigonometric identities.

Accepted Answer

√3 / 2

Answer

-√3 / 2

Answer

1 / 2

Answer

0

Question 11

Determine the range of values for secθ.

Accepted Answer

[-∞, -1] ∪ [1, ∞]

Answer

[-1, 1]

Answer

[-1, 0] ∪ [1, ∞]

Answer

[0, 1] ∪ [1, ∞]

Question 12

Express the Pythagorean identity in trigonometric terms.

Accepted Answer

sin²θ + cos²θ = 1

Answer

tan²θ + cot²θ = 1

Answer

sec²θ - csc²θ = 1

Answer

cot²θ - sin²θ = cos²θ

Question 13

Identify the incorrect half-angle identity.

Accepted Answer

cos(θ/2) = 2cos²θ

Answer

sin(θ/2) = ±√((1 - cosθ)/2)

Answer

cos(θ/2) = ±√((1 + cosθ)/2)

Answer

tan(θ/2) = ±√((1 - cosθ)/(1 + cosθ))

Question 14

If sinθ = 0.6, what is the value of cscθ?

Accepted Answer

5/3

Answer

3/5

Answer

4/3

Answer

3/4

Question 15

Simplify the expression: (sin²θ - cos²θ)/(sinθ + cosθ).

Accepted Answer

tanθ - 1

Answer

sinθ - cosθ

Answer

cosθ - sinθ

Answer

1 - tanθ

Question 16

The value of cos(π/3) is:

Accepted Answer

√3/2

Answer

-1

Answer

0

Answer

1/2

Question 17

Which of the following is the Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

sin(π/2 - θ) = cosθ

Answer

tanθ = sinθ/cosθ

Answer

sec²θ - 1 = tan²θ

Question 18

What is the double-angle identity for cosine?

Accepted Answer

cos2θ = 2cos²θ - 1

Answer

tan2θ = 2tanθ/(1 - tan²θ)

Answer

sin2θ = 2sinθcosθ

Answer

cot2θ = cot²θ - 1

Question 19

What is the value of sin(π/4) using half-angle identities?

Accepted Answer

√2/2

Answer

1/2

Answer

3/4

Answer

1/4

Question 20

Which of the following is NOT a trigonometric identity?

Accepted Answer

sinθ + cosθ = 1

Answer

sin²(θ/2) = (1 - cosθ)/2

Answer

tanθ = sinθ/cosθ

Answer

cos(2θ) = 2sinθcosθ

Question 21

What is the primary purpose of using trigonometric identities?

Accepted Answer

To simplify trigonometric expressions and solve trigonometry problems

Answer

To convert between different trigonometric ratios

Answer

To memorize trigonometric values

Question 22

Which of the following is a common application of trigonometric identities in the field of biology?

Accepted Answer

Modeling periodic phenomena such as circadian rhythms

Answer

Calculating the height of trees

Answer

Solving equations involving angles

Question 23

How can trigonometric identities be used to verify other trigonometric identities?

Accepted Answer

By substituting one identity into another

Answer

By graphically comparing the graphs of the identities

Answer

By solving equations involving the identities

Answer

By using a calculator to evaluate both sides

Question 24

Which of the following options represents the Pythagorean trigonometric identity?

Accepted Answer

sin² θ + cos² θ = 1

Answer

sin θ - cos θ = 1

Answer

sin θ + cos θ = 1

Answer

tan θ + cot θ = 1

Question 25

Identify the double-angle identity for the sine function.

Accepted Answer

sin 2θ = 2sin θ cos θ

Answer

cot 2θ = cot² θ - 1 / 2cot θ

Answer

tan 2θ = 2tan θ / (1 - tan² θ)

Answer

cos 2θ = cos² θ - sin² θ

Question 26

Simplify the trigonometric expression: cos² θ - sin² θ.

Accepted Answer

cos 2θ

Answer

2cos θ sin θ

Answer

tan 2θ

Answer

sin 2θ

Question 27

Which of the following trigonometric identities serves as the foundation for deriving other trigonometric identities?

Accepted Answer

Pythagorean identity

Answer

Sum-to-product identity

Answer

Half-angle identity

Answer

Double-angle identity

Question 28

In proving the double-angle identity for cosine, which step is not included?

Accepted Answer

Substitute sin² θ for cos² θ.

Answer

Use the Pythagorean identity to replace sin² θ.

Answer

Substitute cos 2θ for cos² θ.

Answer

Factor the resulting equation.

Question 29

Which trigonometric identity is used to express the sine or cosine of an angle twice the size of a given angle?

Accepted Answer

Double-angle identity

Answer

Pythagorean identity

Answer

Half-angle identity

Answer

Sum-to-product identity

Question 30

Which of the following is the fundamental Pythagorean identity?

Accepted Answer

sin²θ + cos²θ = 1

Answer

tan²θ + 1 = sec²θ

Answer

sin²θ - cos²θ = 1

Answer

cot²θ - 1 = csc²θ

Question 31

What is the half-angle identity for sine?

Accepted Answer

sin(θ/2) = ±√((1 - cosθ) / 2)

Answer

sin(θ/2) = tan(θ/2)

Answer

sin(θ/2) = cos(θ/2)

Answer

sin(θ/2) = √((1 + cosθ) / 2)

Question 32

Which trigonometric identity is most helpful in solving equations involving tangent?

Accepted Answer

tan²θ + 1 = sec²θ

Answer

cos²θ - sin²θ = 1

Answer

sin²θ + cos²θ = 1

Answer

sin(θ/2) = ±√((1 - cosθ) / 2)

Question 33

Simplify the trigonometric expression: sin²θ - cos²θ

Accepted Answer

-cos(2θ)

Answer

1

Answer

cos²θ - sin²θ

Answer

sin(2θ)

Question 34

Using trigonometric identities, what is the value of cos(π/3)?

Accepted Answer

1/2

Answer

0

Answer

√3/2

Answer

1

Question 35

Which of the following expressions is NOT a trigonometric identity?

Accepted Answer

sinθ + cosθ = 1

Answer

tanθ = sinθ / cosθ

Answer

cotθ = 1 / tanθ

Answer

sin²θ + cos²θ = 1

Question 36

Which trigonometric identity is used to express the sine of an angle half the size of a given angle?

Accepted Answer

Half-angle identity for sine

Answer

Sum-to-product identity

Answer

Double-angle identity for cosine

Answer

Pythagorean identity

Question 37

Simplify the expression: sec²θ - 1

Accepted Answer

tan²θ

Answer

cot²θ

Answer

sin²θ

Answer

cos²θ

Question 38

Which trigonometric identity is represented by the equation cos² θ + sin² θ = 1?

Accepted Answer

Pythagorean Identity

Answer

Double-Angle Identity

Answer

Half-Angle Identity

Answer

Product-to-Sum Identity

Question 39

Simplify the expression: 2sin(θ/2)cos(θ/2)

Accepted Answer

sin(θ)

Answer

cos(θ)

Answer

2sin(θ)

Answer

2cos(θ)

Question 40

Which trigonometric identity is used to find the cosine of the sum of two angles?

Accepted Answer

Cosine Sum Identity

Answer

Sine Sum Identity

Answer

Double-Angle Identity

Answer

Half-Angle Identity

Question 41

Which of the following is an identity?

Accepted Answer

sin⁻¹(sin(x)) = x

Answer

tan⁻¹(tan(x)) = x - π/2

Answer

cos⁻¹(cos(x)) = x + 2nπ

Answer

cot⁻¹(cot(x)) = x + π

Question 42

Which trigonometric identity is used to prove the sum-to-product formula for sine?

Accepted Answer

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

Answer

tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))

Answer

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Question 43

Simplify the expression: (sin x + cos x)² - 1

Accepted Answer

2sin x cos x

Answer

1 - 2sin x cos x

Answer

sin²x + cos²x - 1

Question 44

Which of the following is equivalent to sin 2θ?

Accepted Answer

2sin θ cos θ

Answer

2cos² θ - 1

Answer

1 - 2sin² θ

Answer

cos² θ - sin² θ

Question 45

Given that cos 2θ = 7/25 and 0 < θ < π/2, determine the value of sin θ.

Accepted Answer

3/5

Answer

4/5

Answer

-4/5

Answer

-3/5

Question 46

Express cos 4θ in terms of cos θ.

Accepted Answer

8cos⁴θ - 8cos²θ + 1

Answer

2cos²2θ - 1

Answer

4cos³θ - 3cosθ

Question 47

What is the value of tan(θ/2) if sin θ = 12/13 and π/2 < θ < π?

Accepted Answer

-5/12

Answer

12/5

Answer

5/12

Answer

-12/5

Question 48

Rewrite the expression √[(1 + cos θ)/2] as a single trigonometric function.

Accepted Answer

cos(θ/2)

Answer

sin(θ/2)

Answer

tan(θ/2)

Answer

sec(θ/2)

Question 49

Which of the following is NOT a trigonometric identity?

Accepted Answer

sin(θ + π) = sin θ

Answer

tan θ = sin θ / cos θ

Answer

cos(-θ) = cos θ

Answer

1 + tan²θ = sec²θ

Question 50

A biologist is modeling the growth of a bacterial population using a trigonometric function. The population size, P, at time t hours is given by P(t) = 500sin(πt/12) + 1000. What is the maximum population size?

Accepted Answer

1500

Answer

500

Answer

1000

Question 51

The angle of elevation of the sun is 30°.  A tree casts a shadow 50 feet long on level ground.  What is the height of the tree?

Accepted Answer

25√3 feet

Answer

25 feet

Answer

50√3 feet

Answer

50 feet