Pre-Calculus Mathematics 12 – 6.1 – Trigonometric Identities and Equations

The FUNdamental TRIGonometric Identities

In trigonometry, there are expressions and equations that are true for any given angle. These are called

identities. An infinite number of trigonometric identities exist, and we are going to prove many of these

identities, but we are going to need some basic identities first.

The six basic trig ratios will lead to our first identities

sin θ = cos θ = tan θ =

csc θ = sec θ = cot θ =

And our knowledge of Pythagoras will determine the remaining FUNdamental TRIGonometric Identities

Pythagorean Identities:

2 2sin cos 1

2 21 tan sec 2 21 cot csc

Reciprocal and Quotient Identities:

1sec

cos

1csc

sin

1cot

tan

sintan

cos

coscot

sin

Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on a Trigonometric Expression

Pre-Calculus Mathematics 12 – 6.1 – Trigonometric Identities and Equations

Corollary Identities ( a statement that follows readily from a previous statement)

2 2sin cos 1 2 21 tan sec 2 21 cot csc

Simplifying a Trigonometric Expression

There are many different strategies to simplifying a trigonometric expression. The following examples will

look at the most common types of strategies.

Write as a fraction with a common denominator

Factor as a difference of squares

Change everything to sine and cosine.

Multiply by the conjugate

Now we can use these strategies along with the eight fundamental identities to simplify expressions

Pre-Calculus Mathematics 12 – 6.1 – Trigonometric Identities and Equations

Example 1: Simplify

Example 2: Simplify

Example 3: Simplify

Example 4: Simplify

Example 5: Simplify

Example 6: Simplify

Pre-Calculus Mathematics 12 – 6.1 – Trigonometric Identities and Equations

Restrictions

Just like any algebraic expression, a trigonometric expression cannot have zero in the denominator. We

must consider the exact values that would result in a denominator of zero.

Example 7: Determine the restrictions on for

Practice: Page 213 #2 - 9 (as many as needed)

Pre-Calculus Mathematics 12 – 6.2 – Verifying Trigonometric Identities

Trigonometric Identities

When verifying trigonometric identities, the key is using the rules for algebra as well as the fundamental

trigonometric identities to rewrite and simplify expressions. An identity has been proven when the right

side of the equal sign is the same as the left side of the equal side.

Example 1: Prove the identity:

Example 2: Prove the identity:

Goal: 1. Verify and prove Trigonometric Identities

Pre-Calculus Mathematics 12 – 6.2 – Verifying Trigonometric Identities

Example 3: Prove the identity:

Example 4: Prove the identity:

Practice: Page 271 #1- 26

Pre-Calculus Mathematics 12 – 6.3 –Trigonometric Equations

Trigonometric Equations

A trigonometric equation is different from a trigonometric identity in that the equation is true for some

values of the variable and not all values. When solving a trig equation, we can solve for a specified domain

of values ( usually ) or in general form ( for all potential values)

Example 1: Solve:

√ for

a) b) general form

Example 2: Solve: for a) b) general form

Example 3: Solve: for a) b) general form

Goal: 1. Solve trigonometric equations for conditional statements and general form 2. Solve equation with angles other then θ or x.

Pre-Calculus Mathematics 12 – 6.3 –Trigonometric Equations

Solving trigonometric equations with angles other than θ

Solving a trig equation for angles with coefficients ( 2θ, 3θ, etc) follows very similar steps:

1. Isolate the trig function. eg. sin 2θ

2. Determine the general solution(s) for angle 2θ

3. Determine the general solution(s) for angle θ. (divide by the coefficient)

4. Determine the specific (‘conditional’) solution(s) for the given interval

Example 4: Solve: for a) general form b)

Example 5: Solve: √ for a) general form b)

Example 6: Solve: for a) b) general form

Practice: Page 281 #1 - 5

Pre-Calculus Mathematics 12 – 6.4 –Sum and Difference Identities

Sum and Difference Identities

Identities are not limited to the fundamental identities and single angles. We can also use identities

involving sums and differences. The derivations of these identities are shown on page 287 of your

textbook. We will be looking not at proving these identities but using these identities.

cos cos cos sin sin sin sin cos cos sin

cos cos cos sin sin sin sin cos cos sin

tan tan tan tantan tan

1 tan tan 1 tan tan

Goal: 1. Use sum and difference identities to solve complex trig problems 2. Simplify expressions and prove identities involving sums and differences

Example 1: Find the exact value:

Example 2: Find the exact value:

Example 3: Express as a single function, then evaluate.

Example 4: Express as a single function, then evaluate.

Pre-Calculus Mathematics 12 – 6.4 –Sum and Difference Identities

Example 5: Given angle A in quadrant I and angle B in quadrant II, such that

,

find ( ).

Example 6: Prove:

Practice: Page 292 #1 - 6

Pre-Calculus Mathematics 12 – 6.5– Double-Angle Identities

Double Angle Identities

Using the sum and difference identities, we can determine other trigonometric identities

sin sin cos cos sin

cos cos cos sin sin

tan tan tan tan

tan tan1 tan tan 1 tan tan

Double Angle Identities (same angle)

2 2

2

2

cos2 cos sin

2cos 1

1 2sin

Goal: 1. Identify the double angle trigonometric identities 2. Simplify and prove double-angle trigonometric expressions and equations

sin2 2sin cos

Pre-Calculus Mathematics 12 – 6.5– Double-Angle Identities

Example 3: Prove:

Example 1: Simplify

Example 2: Solve,

Practice: Page 300 #1 – 6 (as many as needed)