Cogito
Trigonometry · Chapter 1 · Lesson 2
The Reciprocal Ratios
Three more names for ratios you already know.
12 problems · about 20 minutes · G-SRT.C.8, F-TF.C.8
What this lesson teaches
The student defines cosecant, secant and cotangent as reciprocals and evaluates them from a right triangle.
- Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent.
- The prefixes cross over: secant pairs with cosine.
- A reciprocal is undefined wherever the original ratio is zero.
Warm Up
Straightforward practice. Get the method working first.
5 problemsWhich ratio is the reciprocal of cosine?
Answer Secant.
Why Secant.
cos θ = 0.5. What is sec θ?
Answer 2
Why 2.
cos θ = 0.25. What is sec θ?
Answer 4
Why 1 ÷ 0.25.
tan θ = 4. What is cot θ, as a decimal?
Answer 0.25
Why 1 ÷ 4.
sin θ = 0.2. What is csc θ?
Answer 5
Why 1 ÷ 0.2.
Build It Up
The same ideas with more to keep track of.
3 problemsOpposite 3, hypotenuse 5. What is csc θ, as a decimal to two places?
Answer 1.67
Why sin θ = 3/5, so csc θ = 5/3.
What is sec 0°?
Answer 1
Why cos 0° = 1.
cot θ = 1. What is tan θ?
Answer 1
Why A number equal to its own reciprocal.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich of these are undefined?
Answer csc 0°; cot 0°
Why A reciprocal breaks where the original ratio is zero, and sin 0° = 0.
Adjacent 8, opposite 6. What is cot θ, as a decimal to two places?
Answer 1.33
Why tan θ = 6/8, so cot θ = 8/6.
The Flip: sin θ = 0.5. What is csc θ?
Answer 2
Why 2.
The Pairs: Match each ratio with its reciprocal.
Answer sine → cosecant; cosine → secant; tangent → cotangent
Why Sine with cosecant, cosine with secant, tangent with cotangent.