Lesson 8 Relationships with Meaning Solidify Understanding

Learning Focus

Examine properties of trigonometric expressions.

What observations can I make about the relationships between trigonometric ratios of the two different reference angles in a right triangle?

How do the properties of a right triangle influence algebraic statements involving trigonometric expressions?

Open Up the Math: Launch, Explore, Discuss

1.

The sine ratio for a $30 °$ angle is $0.5$ . We write this as $\sin (30) = 0.5$ . Use this information to find the missing sides in the right triangle:

2.

a.

You can find the values of the trigonometric ratios on a scientific or graphing calculator.

Find the cosine and sine ratios for $50 °$ on a calculator: $\cos (50) \approx \underset{―}{}$ , $\sin (50) \approx \underset{―}{}$ .

b.

Use this information to find both of the missing sides in the right triangle:

3.

Use your calculator to find the information you need in order to find the desired side in the following right triangles. Record all of your work, including the trigonometric ratios you use, so someone else can follow it.

a.

Find the side adjacent to the $75 °$ angle.

b.

Find the side opposite the $35 °$ angle.

c.

Find the hypotenuse.

4.

Use the information from the given triangle to write the following trigonometric ratios:

$\sin (A) = \frac{opposite}{hypotenuse} = \underset{―}{}$

$\cos (A) = \frac{adjacent}{hypotenuse} = \underset{―}{}$

$\tan (A) = \frac{opposite}{adjacent} = \underset{―}{}$

$\sin (B) = \underset{―}{}$

$\cos (B) = \underset{―}{}$

$\tan (B) = \underset{―}{}$

5.

Use the information from the given triangle to write the following trigonometric ratios:

$\sin (A) = \underset{―}{}$

$\cos (A) = \underset{―}{}$

$\tan (A) = \underset{―}{}$

$\sin (B) = \underset{―}{}$

$\cos (B) = \underset{―}{}$

$\tan (B) = \underset{―}{}$

6.

Use the information from the previous two problems to write observations you notice about the relationships between trigonometric ratios of the two different reference angles in these right triangles.

7.

Do you think these observations will always hold true? Why or why not?

The following is a list of conjectures made by students about right triangles and trigonometric relationships. For each, state whether you think the conjecture is true or false. Justify your answer.

8.

$\cos (A) = \sin (A)$

9.

$\tan (A) = \frac{\sin (A)}{\cos (A)}$

10.

$\sin (A) = \cos (90 ° - A)$

11.

$\cos (A) = \sin (B)$

12.

$\cos (B) = \sin (90 ° - A)$

13.

$\tan (A) = \frac{1}{\tan (B)}$

Note the following convention used to write ${[\sin (A)]}^{2} = \sin^{2} (A)$ .

14.

$\sin^{2} (A) + \cos^{2} (A) = 1$

15.

$1 - \sin^{2} (A) = \cos^{2} (A)$

16.

$\sin^{2} (A) = \sin (A^{2})$

Ready for More?

Given a right triangle with the following trigonometric ratio: $\sin (30 °) = \frac{1}{2}$ , find all of the trigonometric ratios for this triangle. How do you know these values are always going to be true when given this angle?

Takeaways

When given the measures of an acute angle and one side of a right triangle, we can find the other sides by:

An identity is an algebraic equation that is true for all replacements of the variable. When working with trigonometric ratios, some surprising results occur. The following trigonometric identities capture some of the observations we made today for the acute angles in a right triangle:

Vocabulary

complementary angles
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we examined some relationships between trigonometric ratios, such as a relationship between the sine and cosine of complementary angles. We were able to use the properties of a right triangle, including the Pythagorean theorem that describes a relationship between the lengths of the sides, to justify the observations we made today.

Retrieval

1.

Find the circumference and area of the circle.

2.

Find the surface area and volume of the rectangular prism.

3.

What is the difference between the slope of the line segment and the length of the line segment?

Find both measurements.