Ch 7.1: Fundamental Identities and their use

In this section, we will

1. revisit, if necessary, previous encountered identities

2. simplify trig expressions

3. verify trig identities, and

4. write a given expression as a trig expression using a suggestedsubstitution.

Remark: This section, as well as the subsequent ones in Ch. 7,relies heavily on proofs.

Verifying identities

Theorem (Pythagorean identities)

sin2 t + cos2 t = 1

tan2 t + 1 = sec2 t

1 + cot2 t = csc2 t

Why? Let (x , y) be a point on the Unit circle.

Even/Odd Identities

Theorem (Identities due to symmetry)

sin(−t) = − sin(t), csc(−t) = − csc(t)

cos(−t) = cos(t), sec(−t) = sec(t)

tan(−t) = − tan(t), cot(−t) = − cot(t)

For instance,

1. sin(−π/6) =2. cos(−π/6) =3. tan(−π/6) =

Question) Which of the above functions are odd? even?

Proof of theorem

Let (x , y) be a point on the Unit circle.

Example 1Simplify the expression

cos θ + sin θ tan θ.

Example 2Simplify the expression

cot θ +sin θ

1 + cos θ.

Example 3Verify that

2 csc2 t =1

1− cos t+

1

1 + cos t.

Example 4Verify that

tan2 x

1 + sec x=

1− cosx

cos x

Example 6Write

√4− x2 as a trig expression, using the substitution

sin θ = x2 . Assume 0 ≤ θ ≤ π

2 .

Homework for Ch 7.1 (pg. 553)

2, 5, 13, 14, 17, 18, 22, 28, 30, 32.Show work to get credit.