## 1 Fundamental Identities

**Tool:** Identities are true for all values of angle θ
and are

grouped into categories . They may be used to simplify

expressions.

** Reciprocal Identities :**

**Quotient Identities:**

**Even-Odd Identities:**

**Pythagorean Identities:**

**Tool:** Verifying trig identities (see p. 441). An
identity is an

equation that should be true for all values of θ . If it is, then

you can use the identities you already know to simplify both

sides to be equal to each other.

Other suggestions include writing in terms of sines and

cosines, combining fractions over a common denominator,

factoring, and combining like terms .

**Tool:** Disproving false identities takes one single
angle that

shows they don't work. You may have to try more than one if

you try this method .

**Tool:** Sum and difference formulas for cosine maybe
proven

using distance formulas on unit circle using x and y

coordinates (see pp. 446-7)

**Tool:** Cofunction identities by observing that if we
substitute

complementary angles in triangles, we switch our point of

view.

**Tool:** Sum and difference formulas for sine are
proven by

cofunction identities.

**Tool:** Addition and Subtraction identities for
tangent.

## 3 Double and Half Angle Identities

**Tool:** If we set u=v=θ in the sum angle formulas, we

obtain the double angle formulas. Note that double angle

formula for cosine has a couple of equivalent forms based on

the Pythagorean identity.

**Tool:** Half angle identities. These are derived from
the

double angle cosine identities and replacing θ by θ/2

(see p. 462).

**Tool:** Trig equations are true only for certain
values of an

angle ( unlike identities , which are true for all values ).

Always be aware of restrictions. They will tell you where to

look for solutions when you apply inverse trig functions.

**Tool:** Reference triangles can help you find
multiple

solutions of trig equations.

**Tool: **Known solutions and 30-60-90, 45-45-90
triangles can

help solve.

**Tool:** Factoring and finding zeros of factors may
help find

solutions. Think back to when you factored quadratics , but

now we have trig functions.

**Tool:** Trig identities can be used to solve. You may
have to

factor.

## Practice Problems

**Simplifying trig expressions:**

**Verifying Trig identities for all angular values:**

**Proving an equation is not an identitiy:**

**Practice with cosine sum and difference formulas:**

Compute the following exactly (no decimal).

**Cofunction identity practice:**

Given sin(θ)=1/3

What is sec(90−θ)?

**Evaluate the expression exactly (no decimal):**

Hint: recognize the proper identity to simplify

**Evaluate the expression exactly (no decimal):**

Hint: inverse trig functions output angles. Use triangles
to

find those angles.

**(optional)Practice:** You don't really need to
memorize pairs

of sum and difference angle formulas because you can use

even-odd properties to derive one from the other.

Derive the formula for tan(u+v) using even-odd

identities for tangent.

**Evaluating trig functions.**

Let θ be a first quadrant angle such that tan(θ)= 5/12 .

Compute exactly (no decimal).

**Inverse trig.**

Evaluate exactly (no decimal)

**Identity practice.**

Show that

**Half angles.**

If and θ is in first
quadrant, then compute

**Practice solving equations .**

Solve tan(θ)=1 exactly

a) For 0≤θ<90

b) For 0≤θ<360

c) Unrestricted θ

**Practice.**

Solve sin(θ)=0.9563 to two decimal places for

Ans: 73°, 107°

**Solve exactly (no decimal).**

Ans: π/4

**Trig factoring.**