Cogito
Trigonometry · Chapter 4 · Lesson 3
Composing Functions and Inverses
Sometimes the two cancel, and sometimes they do not.
12 problems · about 22 minutes · F-TF.B.6, F-BF.B.4
What this lesson teaches
The student evaluates compositions such as sin(arcsin x) and arcsin(sin x) and explains when they differ.
- sin(arcsin x) returns x whenever x lies between −1 and 1.
- arcsin(sin θ) returns θ only inside the restricted range.
- For mixed compositions, draw the triangle and read off the other ratio.
Warm Up
Straightforward practice. Get the method working first.
5 problemsWhat is arcsin(sin 170°), in degrees?
Answer 10
Why 10°.
Which composition always returns the original input?
Answer sin(arcsin x), for x between −1 and 1.
Why sin(arcsin x). The other is limited by the restricted range.
cos(arccos 0.4)?
Answer 0.4
Why This order cancels.
arcsin(sin 40°), in degrees?
Answer 40
Why 40° is already inside the range.
arcsin(sin 200°), in degrees?
Answer -20
Why sin 200° is negative with reference 20°.
Build It Up
The same ideas with more to keep track of.
3 problemscos(arcsin 0.8)?
Answer 0.6
Why Triangle with opposite 0.8, hypotenuse 1.
sin(arccos 0)?
Answer 1
Why arccos 0 is 90°.
tan(arctan 5)?
Answer 5
Why This order cancels for every real number.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each inverse function with its range.
Answer arcsin → −90° to 90°, endpoints included; arccos → 0° to 180°; arctan → −90° to 90°, endpoints excluded
Why Only arccos covers the upper half of the circle.
arccos(cos 300°), in degrees?
Answer 60
Why cos 300° = 0.5, and arccos lands between 0° and 180°.
The Cancel: What is sin(arcsin 0.3)?
Answer 0.3
Why 0.3.
The Surprise: What is arcsin(sin 150°), in degrees?
Answer 30 degrees
Why 30°.