TOPIC 9 – TRIGONOMETRIC IDENTIES I (PYTHAGOREAN, QUOTIENT, & RECIPROCAL)

In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for

every single value of the occurring variables.

An example is the identity . (On your formula sheet)

Alberta Ed Learning Outcome: Prove Trigonometric Identities using: - Reciprocal Identities - Quotient Identities - Pythagorean Identities - Sum or Difference Identities - Double-Angle Identities. [R, T, V]

Explore 1

Use the / special triangle below to verify the identity

Use your calculator (degree mode) to verify the identity for the non-30/60 angles

Use your graphing calculator to compare the graphs and

Fill out using exact values / the special triangle

Fill out using approximate values / the your calculator Graph and on calc and compare

Verifying the identity

Explore 2 Apply the Pythagorean Formula on the triangle

contained within the unit circle. Verify your

result using the point for .

Trigonometric identities are equalities that involve trigonometric functions that are true for every single value of the occurring variables.

Verifying an does not prove that two expressions form an identity for all values. For this, we must provide a proof.

The Quotient Identities are:

When considering any type of trigonometric identity, we often

must also consider any non-permissible values.

Connect

LS RS

0.0175 0.0175

0.0349 0.0349

0.0523 0.0523

0.0698 0.0698

0.0872 0.0872

0.1045 0.1045

Identities can by verified either numerically (by substituting some angle and evaluating “both sides”) or graphically. (By graphing both sides of an identity)

(same graph)

While the Reciprocal

Identities are:

And the Pythagorean

Identities are:

Example:

Example: State the non-permissible values for

This is Can’t divide by zero!

Undefined where , so

, with another restriction every

In this topic we’ll use the quotient, reciprocal, and

Pythagorean Identities to simplify and prove more

complex identities. (and it’s going to be fun)

1. Given the identity ,

(a) Verify using the angles and (b) Verify graphically. Sketch resulting graphs here

(b) Prove algebraically

(c) State any non-permissible values

2. Simplify each of the following trig expressions to , or 1.

(a) (b) (c)

3. Consider the Pythagorean identity .

(a) If and , use the identity to determine the exact value of .

(b) Isolate and then to develop two counterpart identities

4. Simplify into a single trigonometric expression.

(a) (b) (c) (d)

Practice

Left Side Right Side

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