Lesson 1Reciprocal, Quotient, and Pythagorean Identities

Trigonometric Identities

Lesson 1Reciprocal, Quotient, and Pythagorean Identities

Trigonometric Identity: A trigonometric identity that is true for all permissible values of the variable in the expression on both sides of the equation.

Reciprocal Identities

Quotient Identities

Verify a Potential Identity Numerically and Graphically

Ex. Given :

a) Determine the non-permissible values (NPV’s) of in degrees.

b) Numerically verify that and are solutions of the equation.

c) Use technology to graphically decide whether the equation could be an identity over the domain .

What do you notice?

Use Identities to Simplify Expressions

Ex. Given :

a) Determine the NPV’s of x in radians.

b) Simplify the expression.

Pythagorean Identity

Before we start on the PythagoreanIdentities, let’s consider the following.

A right triangle is drawn within the unit circle ().
A point with coordinates (m, n) is on the circle.


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From the above, we obtain one of the Pythagorean Identities:sin2 A+ cos2 A = 1

... plus its equivalencies:and

Using a similar idea, we have two other Identities:1 + tan2 A = sec2 A

1 + cot2 A = csc2 A

Ex. a) Verify that the equation is true when .

b) Use quotient identities to express the Pythagorean identity as the equivalent identity.

Trigonometric Identities

Lesson 2Sum, Difference, and Double-Angle Identities

Sum Identities:

**NOTE:

Difference Identities:

Question: What if A and B have the same measure?

sin(A + A)cos (A + A)

Double-Angles Identities:

Ex.Write each expression as a single trigonometric function.

a) b)

Ex.Determine an identity for that contains only the cosine ratio.

Ex. Determine an identity for that contains only the sine ratio.

Ex. Consider the expression .

a) What are the NPV’s for the expression?

b) Simplify the expression to one of the three primary trigonometric functions.

c) Verify your answer from part b), for the interval , using technology.

Ex. Determine the exact value for each expression.

a)

b)

Trigonometric Identities

Lesson 3Proving Identities

To prove that an identity is true for all permissible values, it is necessary to express both sides of the identity in equivalent forms.

Ex. (Verify vs. Prove)

a) Verify that for some values of x. Determine the NPV’s of x.

b) Prove that for all permissible values of x.

Ex. Prove that is an identity for all permissible values of x.

We use our Trigonometric Identities to prove any given identity!

Here are some additional strategies:

1. Manipulate the more complicated side.

2. Rewrite all trig ratios into sin and cos only.

3. Multiply the numerator & denominator by the conjugate of an expression.

4. Factor to simplify expressions.

**NOTE:Work on the Left Side (LS) and the Right Side (RS) separately ...

... nocrossmultiplying!

Ex. Prove that is an identity for all permissible values of x.

Ex. Prove that is an identity for all permissible values of x.

Trigonometric Identities

Lesson 4Solving Trigonometric Equations Using Identities

With the identities, we are able to solve more trigonometric equations. ALWAYS CHECK YOUR ANSWER(S) FOR NON-PERMISSABLE VALUES (NPVs)!

Ex. Solve each equation algebraically over the domain .

a)

b)

Ex. a) Solve the equation algebraically in the domain .

b) Verify your answer graphically.

Ex. Solve the equation algebraically. Give the generalsolution (in radians).

Ex. Algebraically solve . Give general solutions expressed in radians.