Lesson 5 Maintaining Your Identity Solidify Understanding

Learning Focus

Derive and justify trigonometric identities.

Statements like the distributive property of multiplication over addition or the associative property of addition help us work algebraically to change the form of expressions into more useful forms. What are some of the properties of trigonometric expressions that might help us change the forms of trigonometric expressions in useful ways?

Open Up the Math: Launch, Explore, Discuss

Right triangles and the unit circle provide images that can be used to derive, explain, and justify a variety of trigonometric identities.

1.

For example, how might the right triangle diagram help you justify why the following identity is true for all angles $θ$ between $0 °$ and $90 °$ ?

$\sin (θ) = \cos (90 ° - θ)$

Since we have extended our definition of the sine to include angles of rotation, rather than just the acute angles in a right triangle, we might wonder if this identity is true for all angles $θ$ , not just those that measure between $0 °$ and $90 °$ .

A version of this identity that uses radian rather than degree measure would look like this:

$\sin (θ) = \cos (\frac{π}{2} - θ) .$

2.

Justify why this identity is true for all angles $θ$ .

Fundamental Trigonometric Identities

There are a variety of ways to discover, explore, and explain trigonometric identities. For example,

In the previous lesson we used the unit circle to show $\cos (- θ) = \cos (θ)$

We can use the angle of rotation definition of the tangent, $\tan (θ) = \frac{y}{x}$ , to show that $\tan (θ) = \frac{\sin (θ)}{\cos (θ)}$ since $y = \sin (θ)$ and $x = \cos (θ)$ on the unit circle.
You can also use graphs to show that two trigonometric expressions are equivalent. For example, we have already observed that the graph of $y = \sin (- θ)$ is the reflection across the horizontal axis of the graph of $y = \sin (θ)$ , which leads to the identity $\sin (- θ) = - \sin (θ)$ .

These strategies are preferred over using a right triangle to justify a trigonometric identity, since they show the identity is true for all angles of rotation, not just the acute angles of a right triangle.

3.

Here is an important identity known as the Pythagorean identity:

$\sin^{2} θ + \cos^{2} θ = 1$ . [Note: This is the conventional notation for $(\sin {θ)}^{2} + (\cos {θ)}^{2} = 1$ ]

a.

Use a right triangle to show the Pythagorean identity is true for all acute angles.

b.

Use another method to show the identity is true for all angles of rotation.

4.

Use graphs or a unit circle to help you form a conjecture for how to complete the following statements as trigonometric identities.

a.

How might you use the other representation to find additional supporting evidence that your conjectures are true?

b.

$\sin (π - θ) =$ and $\cos (π - θ) =$

c.

$\sin (π + θ) =$ and $\cos (π + θ) =$

d.

$\sin (2 π - θ) =$ and $\cos (2 π - θ) =$

5.

We can use algebra, along with some fundamental trigonometric identities, to prove other identities. For example, how can you use algebra and the identities listed above to prove the following identities?

a.

$\tan (- θ) = - \tan (θ)$

b.

$\tan (π + θ) = \tan (θ)$

Pause and Reflect

6.

Suppose you know $\sin (θ) = - 0.75$ and $π < θ < \frac{3 π}{2}$ . Use the Pythagorean identity to find the following:

a.

$\cos (θ) =$

b.

$\tan (θ) =$

Ready for More?

Is $\sin (α + β) = \sin α + \sin β$ a true statement (i.e., an identity) or a false statement? What makes you say so?

Takeaways

Trigonometric Identities are

The Fundamental Trigonometric Identities that will be useful in future work include the following:

Vocabulary

trigonometric identities
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we identified and explained some fundamental trigonometric identities—trigonometric statements that are true for all angles. Trigonometric identities will allow us to change the form of a trigonometric expression, when needed. One of the identities, $\sin^{2} θ + \cos^{2} θ = 1$ , looks complicated and may seem like it shouldn’t be true, but it can be verified using the Pythagorean theorem.

Retrieval

1.

In the diagram, triangle $A B C$ is a right triangle.

Point $B$ is a point on the circle and is described by the rectangular coordinates $(6, 8)$ .
$s$ is the length of the arc subtended by angle $θ$ .
$r$ is the radius of circle $A$ .

a.

Find $r$ .

$r =$

b.

Find $θ$ to the nearest thousandth of a radian. ( $3$ decimal places)

$θ =$

c.

Find $s$ by using the formula $s = r θ$ .

$s =$

d.

Describe point $B$ using the coordinates $(r, θ)$ .

e.

Describe point $B$ using the radius and arc length $(r, s)$ .

2.

Find two solutions in degrees and two solutions in radians. $(0 ° \leq θ \leq 360 °)$ and $(0 \leq θ \leq 2 π)$ . Do NOT use a calculator.

$\sin θ = - \frac{\sqrt{3}}{2}$