M5.3.5 For the adult
Identities
Which trig facts are true for every angle, and how do I prove one?
What this chapter teaches
An identity is true for every value of x. The key ones are sin²x + cos²x ≡ 1, from Pythagoras on the unit circle, and tan x ≡ sin x ÷ cos x. To prove an identity, change one side into the other with steps that are always valid, or change both sides to the same expression. Never start by assuming the result. A single value that fails shows that something is not an identity.
Three questions to ask
There is no one right wording. Listen for the idea.
- Why does sin²x + cos²x equal 1 for every angle?
- Why is sin x + cos x = 1 not an identity?
- Why is it wrong to prove an identity by starting from it and rearranging?
After the chapter, the learner can
- Use sin²x + cos²x = 1 to simplify an expression
- Write tan x as sin x divided by cos x
- Prove a simple trig identity
What they copy into their notebook
sin²x + cos²x ≡ 1. tan x ≡ sin x ÷ cos x. Prove: work on one side, tan into sin and cos, common denominators, multiply by 1 + sin x. One counterexample disproves.