Reciprocal and inverse trig functions

What this chapter teaches

The reciprocal functions are sec x = 1/cos x, cosec x = 1/sin x and cot x = 1/tan x. They have asymptotes where the bottom is 0, and sec and cosec never take values between −1 and 1. Dividing sin²x + cos²x = 1 by cos²x gives 1 + tan²x = sec²x. The inverse functions arcsin, arccos and arctan undo sin, cos and tan on a stretch where each is one to one. Their answers always lie in a fixed range.

Three questions to ask

There is no one right wording. Listen for the idea.

1. What is sec 60°, and why can sec never be 0.5?
2. Why does sin x need to be cut down to −π/2 ≤ x ≤ π/2 before it has an inverse?
3. Is sin⁻¹ 0.5 the same as 1 ÷ sin 0.5?

After the chapter, the learner can

• Sketch the graphs of sec, cosec and cot
• Use 1 + tan²x = sec²x
• Give the domain and range of arcsin x