# Eureka Math Algebra 2 Module 2 Lesson 15 Answer Key

## Engage NY Eureka Math Algebra 2 Module 2 Lesson 15 Answer Key

### Eureka Math Algebra 2 Module 2 Lesson 15 Exercise Answer Key

Exercise 1.
Recall the Pythagorean identity sin2(θ) + cos2(θ) = 1, where θ is any real number.
a. Find sin(x), given cos(x) = $$\frac{3}{5}$$, for $$\frac{\pi}{2}$$ < x < 0.

b. Find tan(y), given cos(y) = –$$\frac{5}{13}$$ for $$\frac{\pi}{2}$$ < y < π.

c. Write tan(z) in terms of cos(z), for π < z < $$\frac{3 \pi}{2}$$.

Exercise 2.
Use the Pythagorean identity to do the following:
a. Rewrite the expression cos(θ) sin2(θ) – cos(θ) in terms of a single trigonometric function. State the resulting identity.
cos(θ) sin2(θ) – cos(θ) = cos(θ) (sin2(θ) – 1)
= cos(θ) (- cos2(θ))
= – cos3(θ)
Therefore, cos(θ) sin2(θ) – cos(θ) = – cos3(θ) for all real numbers θ.

b. Rewrite the expression (1 – cos2(θ)) csc(θ) in terms of a single trigonometric function. State the resulting identity.
(1 – cos2(θ)) csc(θ) = sin2(θ) csc(θ)
= sin2(θ) $$\frac{1}{\sin (\theta)}$$
= sin(θ)
Therefore, (1 – cos2(θ)) csc(θ) = sin(θ) for θ ≠ kπ,for all integers k.

c. Find all solutions to the equation 2 sin2(θ) = 2 + cos(θ) in the interval (θ, 2π). Draw a unit circle that shows the solutions.
2 sin2(θ) = 2 + cos(θ)
2(1 – cos2(θ)) = 2 + cos(θ)
2 – 2cos2(θ) = 2 + cos θ
– 2cos2(θ) – cos(θ) = θ
2 cos2(θ) + cos(2) = θ
cos(θ) (2 cos(θ) + 1) = θ
Therefore, cos(θ) = 0, or cos(θ) = –$$\frac{1}{2}$$
See the unit circle on the right, which shows the four points where cos(θ) = 0, or cos(θ) = –$$\frac{1}{2}$$.
In the interval (θ, 2π), cos(θ) = 0 only if θ = $$\frac{\pi}{2}$$, or
θ = $$\frac{3 \pi}{2}$$. Also, cos(θ) = –$$\frac{1}{2}$$ only if θ = $$\frac{2 \pi}{3}$$, or θ = $$\frac{4 \pi}{3}$$ .
Therefore, the solutions of the equations in the interval (0, 2π) are $$\frac{\pi}{2}, \frac{3 \pi}{2}, \frac{2 \pi}{3}$$, and $$\frac{4 \pi}{3}$$.

Exercise 3.
Which of the following statements are identities? If a statement is an identity, specify the values of x where the equation holds.
a. sin(x + 2π) = sin(x) where the functions on both sides are defined.
This is an identity defined for all real numbers.

b. sec(x) = 1 where the functions on both sides are defined.
This is not an identity. The functions are not equivalent for all real numbers. For example, although sec(0) = 1, sec$$\left(\frac{\pi}{4}\right)$$ = √2. The functions have equal values only when x is an integer multiple of 2π. Additionally, the ranges are different. The range of f(x) = sec(x) is all real numbers y such that y ≤ – 1 or y ≥ 1, whereas the range of g(x) = 1 is the single number 1.

c. sin(-x) = sin(x) where the functions on both sides are defined.
This is not an identity; this statement is only true when sin(x) = 0, which happens only at integer multiples of π.

d. 1 + tan2(x) = sec2(x) where the functions on both sides are defined.
This is an identity. The functions on either side are defined for θ ≠ $$\frac{\pi}{2}$$ + kπ, for all integers k.

e. sin ($$\frac{\pi}{2}$$ – x) = cos(x) where the functions on both sides are defined.
This is an identity defined for all real numbers.

f. sin2(x) = tan2(x) for all real x.
This is not an identity. The equation sin2(x) = tan2(x) is only true where sin2(x) = $$\frac{\sin ^{2}(x)}{\cos ^{2}(x)}$$, so cos2(x) = 1, and then cos(x) = 1, or cos(x) = – 1, which gives x = πk, for all integers k. For all other values of x, the functions on the two sides are not equal. Moreover, tan2(x) is defined only for θ ≠ $$\frac{\pi}{2}$$ + kπ, for all integers k, whereas sin2(x) is defined for all real numbers.

Another argument for why this statement is not an identity is that sin2$$\left(\frac{\pi}{4}\right)=\left(\frac{\sqrt{2}}{2}\right)^{2}=\frac{1}{2}$$ , but tan2$$\left(\frac{\pi}{4}\right)$$ = 12 = 1, and 1 ≠ $$\frac{1}{2}$$ therefore, the statement is not true for all values of x.

### Eureka Math Algebra 2 Module 2 Lesson 15 Problem Set Answer Key

Question 1.
Which of the following statements are trigonometric identities? Graph the functions on each side of the equation.
a. tan(x) = $$\frac{\sin (x)}{\cos (x)}$$ where the functions on both sides are defined.

This is an identity that is defined for x ≠ $$\frac{\pi}{2}$$ + kπ, for all integers k. See the identical graphs above.

b. cos2(x) = 1 + sin(x) where the functions on both sides are defined.
This is not an identity. For example, when x = 0, the left side of the equation is 1, and the right side is also 1.
But when x = $$\frac{\pi}{2}$$, the left side is 0, and the right side is 2. The graphs below are clearly different.

c. cos ($$\frac{\pi}{2}$$ – x) = sin(x) where the functions on both sides are defined.
This is an identity that is defined for all real numbers x. See the identical graphs below.

Question 2.
Determine the domain of the following trigonometric identities:
a. cot(x) = $$\frac{\cos (x)}{\sin (x)}$$ where the functions on both sides are defined.
This identity is defined only for x ≠ kπ, for all integers k.

b. cos(-u) = cos (u) where the functions on both sides are defined.
This identity is defined for all real numbers u.

c. sec(y) = $$\frac{1}{\cos (y)}$$ where the functions on both sides are defined.
This identity is defined for y ≠ $$\frac{\pi}{2}$$ + kπ, for all integers k.

Question 3.
Rewrite sin(x)cos2(x) – sin(x) as an expression containing a single term.
sin(x) – sin(x)cos2(x) = sin(x)(1 – cos2(x))
= sin(x)sin2(x)
= sin3(x)

Question 4.
Suppose 0 < θ < $$\frac{\pi}{2}$$ and sin(θ) = $$\frac{1}{\sqrt{3}}$$. What is the value of cos(θ)?
cos(θ) = $$\frac{\sqrt{6}}{3}$$

Question 5.
If cos(θ) = –$$\frac{1}{\sqrt{5}}$$ what are possible values of sin(θ)?
Either sin(θ) = $$\frac{2}{\sqrt{5}}$$ or sin(θ) = –$$\frac{2}{\sqrt{5}}$$

Question 6.
Use the Pythagorean identity sin2(θ) + cos2(θ) = 1, where θ is any real number, to find the following:
a. cos(θ), given sin(θ) = $$\frac{5}{13}$$, for $$\frac{\pi}{2}$$ < θ < π.

b. tan(x), given cos(x) = –$$\frac{1}{\sqrt{2}}$$, for π < x < $$\frac{3 \pi}{2}$$.

Question 7.
The three identities below are all called Pythagorean identities. The second and third follow from the first, as you saw in Example 1 and the Exit Ticket.
a. For which values of θ are each of these identities defined?
i. sin2(θ) + cos2(θ) = 1, where the functions on both sides are defined.
Defined for any real number θ.

ii. tan2(θ) + 1 = sec2(θ), where the functions on both sides are defined.
Defined for real numbers θ such that θ ≠ $$\frac{\pi}{2}$$ + kπ, for all integers k.

iii. 1 + cot2(θ) = csc2(θ), where the functions on both sides are defined.
Defined for real numbers θ such that θ ≠ kπ, for all integers k.

b. For which of the three identities is 0 in the domain of validity?
Identities i and ii

c. For which of the three identities is $$\frac{\pi}{2}$$ in the domain of validity?
Identities i and iii

d. For which of the three identities is –$$\frac{\pi}{4}$$ in the domain of validity?
Identities i, ii, and iii

### Eureka Math Algebra 2 Module 2 Lesson 15 Exit Ticket Answer Key

April claims that 1 + $$\frac{\cos ^{2}(\theta)}{\sin ^{2}(\theta)}$$ = $$\frac{1}{\sin ^{2}(\theta)}$$ is an identity for all real numbers θ that follows from the Pythagorean identity.

a. For which values of θ are the two functions f(θ) = 1 + $$\frac{\cos ^{2}(\theta)}{\sin ^{2}(\theta)}$$ and g(θ) = $$\frac{1}{\sin ^{2}(\theta)}$$ defined?
Both functions contain sin(θ) in the denominator, so they are undefined if sin(θ) = 0. Thus, the two functions f and g are defined when θ ≠ kπ, for all integers k.

b. Show that the equation 1 + $$\frac{\cos ^{2}(\theta)}{\sin ^{2}(\theta)}$$ = $$\frac{1}{\sin ^{2}(\theta)}$$ follows from the Pythagorean identity.
d. Write the equation 1 + $$\frac{\cos ^{2}(\theta)}{\sin ^{2}(\theta)}$$ = $$\frac{1}{\sin ^{2}(\theta)}$$in terms of other trigonometric functions, and state the resulting identity.