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Reciprocal Trig Functions: csc, sec, cot

Grade 10·20 problems·~30 min·ACT Math·topic·act-geo-trig-recip
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A

Recall / Warm-Up

1.

In a right triangle, sin⁡(θ)\sin(\theta) equals which ratio?

2.

What is the value of cos⁡(60°)\cos(60\degree)?

3.

In the right triangle shown, which expression equals the tangent of angle A?

B

Fluency Practice

1.

Which function is the reciprocal of sin⁡(θ)\sin(\theta)?

2.

In a right triangle, if the side opposite angle θ\theta is 5 and the hypotenuse is 13, what is csc⁡(θ)\csc(\theta)?

3.

What is csc⁡(30°)\csc(30\degree)?

4.

What is sec⁡(30°)\sec(30\degree)? Express your answer in rationalized form.

5.

What is cot⁡(60°)\cot(60\degree)? Express your answer in rationalized form.

C

Varied Practice

1.

In the right triangle shown, what is the value of sec⁡(A)\sec(A)?

2.

What is sec⁡(45°)\sec(45\degree)? Express your answer in simplified radical form.

3.

Which expression is equal to csc⁡(θ)\csc(\theta)?

4.

Simplify sin⁡(θ)⋅csc⁡(θ)+cos⁡(θ)⋅sec⁡(θ)\sin(\theta) \cdot \csc(\theta) + \cos(\theta) \cdot \sec(\theta). The result is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

result:
5.

Which of the following is undefined?

D

Word Problems / Application

1.

On the ACT, you see this expression:

sec⁡(θ)⋅cos⁡(θ)−cot⁡(θ)⋅tan⁡(θ)\sec(\theta) \cdot \cos(\theta) - \cot(\theta) \cdot \tan(\theta)

What does this expression simplify to?

2.

An ACT problem states: "If sin⁡(θ)=35\sin(\theta) = \frac{3}{5} and θ\theta is an acute angle, what is csc⁡(θ)\csc(\theta)?"

Which answer choice is correct?

3.

On the ACT, you encounter: "Simplify cot⁡(θ)⋅sin⁡(θ)\cot(\theta) \cdot \sin(\theta)."

Which single trig function does this expression equal?

E

Error Analysis

1.

Marcus solved this problem:

"What is sec(30 degrees)?"

Marcus's work:

  1. The reciprocal of sec is sin.
  2. sin⁡(30°)=12\sin(30\degree) = \frac{1}{2}
  3. sec⁡(30°)=1sin⁡(30°)=11/2=2\sec(30\degree) = \frac{1}{\sin(30\degree)} = \frac{1}{1/2} = 2

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

2.

Priya solved this problem:

"Find csc(60 degrees)."

Priya's work:

  1. csc⁡(60°)=sin⁡−1(60°)\csc(60\degree) = \sin^{-1}(60\degree)
  2. Using a calculator: sin⁡−1(60)\sin^{-1}(60) gives an error (domain issue)
  3. "The answer is undefined."

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

F

Challenge / Extension

1.

If sec⁡(θ)=3\sec(\theta) = 3, what is cos⁡2(θ)+sin⁡2(θ)+tan⁡2(θ)\cos^2(\theta) + \sin^2(\theta) + \tan^2(\theta)?

2.

Explain why csc⁡(θ)\csc(\theta) is always greater than or equal to 1 or less than or equal to −1-1 for all angles where it is defined. What does this tell you about the range of cosecant?

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