Trigonometric Identities

Reciprocal, Quotient, and Pythagorean Identities

A trigonometric _________________ is a trigonometric equation that is ___________ for all permissible values of the variables in the expressions on both sides of the equation.

Verifying an Identity Graphically:

To check to see if an equation is an identitiy, graph __________ sides of the equation. If both graphs are ______________________, then the equation is an idenitity for that _____________. In other words, the ____________-hand side is equal to the ________________- hand side.

Example: Is sin cos tanx x x= an identity? Graph:

Reciprocal Idenitities: Quotient Identitites:

To determine the non-permissible values, assess each trigonometric funtion ________________________ and examine ____________________ that may have non-permissible values.

Example 1: Given the following equation: tansecsin

θθθ

=

a.) Determine the non-permissible values in radians and degrees.

b.) Numerically verify that 60θ = ° and 4πθ = are solutions of the equation.

Using Identities to Simplify Expressions Examples: Simplify each of the following and state any non-permissible values.

1.) cos tanx x

2.) cotcsc cos

xx x

Pythagorean Identities:

1.) Recall the equation of a unit circle:

2.) If the Pythagorean idenitity is divided by 2cos θ , the result is: 3.) If the Pythagorean idenitity is divided by 2sin θ , the result is:

Example: Verify that the equation 2 2cot 1 cscx x+ = is true when 6

x π= .

Proving Identities

When proving an idenitity, you are showing that the identity is true for all permissible values. One or both sides of the identity must be algebraically manipulated into an equivalent form to match the other side. You must simplify each side independently and therefore you cannot put ____________ signs across the 2 sides throughout your proof. Example 1: Prove the following identities:

1.) cot sin cosθ θ θ=

2.) 21 sin sin cos cotx x x x− =

Helpful Hints: Some common operations when proving identities are:

1. Multiplying/Dividing rational expressions

Ex1. 2 2

1 1cos sinθ θ

Ex2:

cossinsin

θθθ

2. Adding/Subtracting rational expressions

Ex. 2 2

1 1cos sinθ θ

+

3. Factoring

Ex1. 3 2sin sin cosθ θ θ+ Ex2: 4 4sin cosx x−

4. Multiplying by the conjugate

Ex. 1

1 sinθ+

Example 2: Prove each of the following:

1. 2tan cot 2sin 1tan cot

θ θ θθ θ−

= −+

2. cos tan sin secθ θ θ θ+ =

3. sin cos sec csc 1θ θ θ θ =

4. 2

22

1 tan csctan

θ θθ

+=

Sum, Difference, and Double-Angle Identities

Sum Formulae:

( )( )

( )

sin sin cos cos sin

cos cos cos sin sintan tantan

1 tan tan

α β α β α β

α β α β α βα βα βα β

+ = +

+ = −

++ =

−

Difference Formula:

( )( )

( )

sin sin cos cos sin

cos cos cos sin sintan tantan

1 tan tan

α β α β α β

α β α β α βα βα βα β

− = −

− = +

−− =

+

Double Angle Identities

2 2

2

sin 2 2sin coscos 2 cos sin

2 tantan 21 tan

α α α

α α αααα

=

= −

=−

The double angle idenitity for cosine can also be simplified to:

1.)

2.)

Simplifying Expressions Example 1: Write each expression as a single trigonometric function and evaluate. State exact answers if possible, otherwise state approximate answers to 4 decimal places.

a.) sin 48 cos17 cos 48 sin17° ° − ° °

b.) 2 2cos sin3 3π π−

Example 2: Given the following expression: 1 cos 2sin 2

xx

− :

a.) What are the permissible values?

b.) Simplify to a single primary trigonometric function.

Example 3: Simplify cos2

xπ +

to a single trigonometric function.

Proofs with Sum and Difference Identities

Example 4: Prove 1 3sin sin cos3 2 2

x x xπ + = +

.

Example 5: Prove that ( ) ( )sin sin 2cos sinα β α β α β+ − − =

Using Identities to Determine Exact Values We can use these identities to find exact values that are not covered on the special triangles. Example 1: Determine the exact value for each of the following:

a.) cos12π

b.) sin195°

c.) tan105°

Sum and Difference and Double Angle Identities continued:

Example 2: Angle θ is in quadrant III and 3cos5

θ = − . Determine an exact value for

each of the following:

a.) cos 2θ

b.) sin 2θ

c.) ( )cos π θ+

Example 3: Given 3sin5

A = − , where 3 22

Aπ π≤ ≤ and 40cos41

B = where 3 22

Bπ π≤ ≤ ,

find ( )cos A B+ .

Proving Identities Using Double Angles

Example 1: Prove that 1 cos 2tansin 2

xxx

−= is an identity for all pemissible values of x.

Example 2: Prove the identity cos 2 coscot cscsin 2 sin

x xx xx x−

− =+

for all permissible values of x.

**Note: _________________ an identitity using a specific value validates that it is true for that value only. ________________ an identitity is done algebraically and validates the identity for all permissible values of the variable.

Solving Trigonometric Equations using Identities

To solve some trigonometric equations, you need to make substitutions using trigonometric identities. This often involves ensuring that the equation is expressed in terms of one trigonometric function. Example 1: Solve each equation algebraically over the domain 0 2x π≤ ≤ , and state any restrictions.

a.) cos 2 1 cos 0x x+ − =

b.) 21 cos 3sin 2x x− = −

Example 2: Solve each of the following giving general solutions expressed in radians.

a.) sin 2 2 cosx x=

b.) 2sin 7 3cscx x= −