Exercises: Pythagorean identity and basic trig identities - ACT Math

A

Recall / Warm-Up

1.

In a right triangle, which ratio defines $\sin\theta$ ?

A.

opposite / hypotenuse

B.

adjacent / hypotenuse

C.

opposite / adjacent

D.

hypotenuse / opposite

2.

A point on the unit circle at angle θ has coordinates (x, y). Which statement is correct?

A.

$x = \sin\theta$ and $y = \cos\theta$

B.

$x = \cos\theta$ and $y = \sin\theta$

C.

$x = \tan\theta$ and $y = \sec\theta$

D.

$x = \sec\theta$ and $y = \csc\theta$

3.

In which quadrants is cosine negative?

A.

Quadrants I and II

B.

Quadrants II and III

C.

Quadrants III and IV

D.

Quadrants I and IV

B

Fluency Practice

1.

If $\sin\theta = \frac{3}{5}$ and $\theta$ is in Quadrant I, what is $\cos\theta$ ? Express your answer as a fraction.

2.

If $\cos\theta = \frac{5}{13}$ and $\theta$ is in Quadrant I, what is $\sin\theta$ ? Express your answer as a fraction.

3.

If $\tan\theta = 2$ and $\theta$ is in Quadrant I, what is $\sec\theta$ ? Express your answer in simplified radical form.

4.

Which expression is equivalent to $\frac{\sin\theta}{\cos\theta}$ ?

A.

$\sec\theta$

B.

$\csc\theta$

C.

$\tan\theta$

D.

$\cot\theta$

5.

Simplify: $\sin\theta \cdot \sec\theta$ .

A.

$\cos\theta$

B.

$\tan\theta$

C.

$\csc\theta$

D.

1

C

Varied Practice

1.

If $\sin\theta = \frac{3}{5}$ and $\theta$ is in Quadrant II, what is $\cos\theta$ ? Express your answer as a fraction.

2.

Which of the following is the most fully simplified equivalent of $\sin^2\theta + \cos^2\theta + \tan^2\theta$ , expressed using a single trigonometric function?

A.

$1 + \tan^2\theta$

B.

$\sec^2\theta$

C.

$\csc^2\theta$

D.

$2 + \tan^2\theta$

3.

Complete the identity: $1 + \cot^2\theta$ = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

right side:

4.

If $\cot\theta = 3$ and $\theta$ is in Quadrant I, what is $\csc\theta$ ? Express your answer in simplified radical form.

5.

A student claims that $\sin\theta + \cos\theta = 1$ for all angles $\theta$ . Which value of $\theta$ disproves this claim?

A.

$\theta = 0\degree$

B.

$\theta = 45\degree$

C.

$\theta = 90\degree$

D.

$\theta = 360\degree$

D

Word Problems / Application

1.

A surveyor measures an angle $\theta$ from horizontal to the top of a building. She determines that $\cos\theta = -\frac{5}{13}$ and $\theta$ is in Quadrant III.

1.

What is $\sin\theta$ ? Express your answer as a fraction.

2.

What is $\tan\theta$ ? Express your answer as a fraction.

2.

An engineer needs to verify that a calculated angle satisfies a structural constraint. She knows that $\sec\theta = -3$ and $\theta$ is in Quadrant II.

What is $\tan\theta$ ? Express your answer in simplified radical form.

A.

$-2\sqrt{2}$

B.

$2\sqrt{2}$

C.

$-\sqrt{8}$

D.

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

E

Error Analysis

1.

Marcus solved this problem:

"If $\sin\theta = \frac{4}{5}$ and $\theta$ is in Quadrant II, find $\cos\theta$ ."

Marcus's work:

$\sin^2\theta + \cos^2\theta = 1$
$\frac{16}{25} + \cos^2\theta = 1$
$\cos^2\theta = \frac{9}{25}$
$\cos\theta = \frac{3}{5}$

What error did Marcus make, and what is the correct answer?

A.

Marcus forgot that cosine is negative in Quadrant II. The correct answer is $\cos\theta = -\frac{3}{5}$ .

B.

Marcus used the wrong identity. He should have used $\tan^2\theta + 1 = \sec^2\theta$ .

C.

Marcus squared $\frac{4}{5}$ incorrectly. $\left(\frac{4}{5}\right)^2 = \frac{8}{10}$ , not $\frac{16}{25}$ .

D.

Marcus's answer of $\frac{3}{5}$ is correct.

2.

Priya simplified the expression $\cos^2\theta \cdot \tan^2\theta$ :

Priya's work:

Replace $\tan\theta$ with $\frac{1}{\cos\theta}$
$\cos^2\theta \cdot \frac{1}{\cos^2\theta} = 1$

What error did Priya make, and what is the correct simplification?

A.

Priya confused $\tan\theta$ with $\sec\theta$ . The correct simplification is $\sin^2\theta$ .

B.

Priya forgot to square the tangent. The correct answer is $\cos\theta \cdot \tan\theta$ .

C.

Priya's answer of 1 is correct.

D.

Priya should have gotten $\cos^2\theta$ because the tangents cancel.

F

Challenge / Extension

1.

If $\sin\theta = \frac{7}{25}$ and $\theta$ is in Quadrant II, find the exact values of all six trigonometric functions of $\theta$ .

2.

Simplify: $\frac{\tan^2\theta}{\sec\theta + 1}$ .

A.

$\sec\theta - 1$

B.

$\sec\theta + 1$

C.

$\tan\theta - 1$

D.