Writing Trigonometric Ratios

Warm-Up

A trigonometric ratio is a ratio of two sides of a right triangle. We can use trigonometric ratios to relate the angles of a triangle to the lengths of the triangle’s sides. In a right triangle, three relationships exist which relate the angles of a triangle to the measures of its sides.

The word opposite means _________________________________________________

The word adjacent means _________________________________________________

The hypotenuse is _______________________________________________________

sine (sin) of Angle x = SideOpposite ¿ Angle x ¿

Hypotenuse

cosine (cos) of Angle x = Side Adjacent ¿ Angle x ¿

Hypotenuse

tangent (tan) of Angle x = SideOpposite ¿ Angle x ¿Side Adjacent ¿

Angle x ¿

We write:

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sin x=opp xhyp

cos x=adj xhyp

tan x=opp xadj x

Remember this!---------------------------------------->

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Example

For the given triangle, express as a fraction:

a) sin A

b) cos A

c) tan A

For the given triangle, express as a fraction:

d) sin B

e) cos B

f) tan B

Express each of the above as a decimal rounded to the nearest ten-thousandth:

sin B = _______________ cos B = ___________________ tan B = _______________

Exercise

1) For the given triangle, express as a fraction:

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Remember to label all sides first:OPPOSITE, ADJACENT, HYPOTENUSE

Sides OPPOSITE and ADJACENT will change depending on which angle you’re talking about.

The HYPOTENUSE will always be across from the right angle. So label it first!

a) sin K

b) cos K

c) tan K

d) sin J

e) cos J

f) tan J

2) For the given triangle, express as a fraction and a decimal rounded to the nearest ten-thousandth:

a) cos L

b) tan L

c) sin L

d) sin M

e) tan M

f) cos M

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Finding Values on the Calculator

The sine, cosine, and tangent of any angle is a known value that can be found using any scientific calculator.

Examples Using your calculator, find the value of each to the nearest ten-thousandth:

sin 38 ° _______________ cos38 ° _______________tan38 ° ___________________

Exercise Using your calculator, find the value of each to the nearest ten-thousandth:

tan22 ° _______________ cos17° _______________tan31 ° ___________________

sin 80 ° _______________ sin 42 ° _______________cos60 ° ___________________

sin 45 ° _______________tan65 ° _______________cos 44 ° ___________________

Lesson Summary

The Three Trigonometric Ratios

sin x=opp xhyp

cos x=adj xhyp

tan x=opp xadj x

Exit ticket

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Homework Express each ratio indicated as a fraction.

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Finding a Missing Side Length Using the Trigonometric Ratios

Warm-Up

The trigonometric ratios can help us to solve for a missing side in a right triangle.

sin x=opp xhyp

cos x=adj xhyp

tan x=opp xadj x

Example #1

Find, to the nearest tenth:

a) AB

b) AC

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Example #2

Find, to the nearest tenth:

a) MP

b) NM

Exercise

1) Find, to the nearest tenth:

a) XY

b) YZ

2) Find, to the nearest tenth:

a) DE

b) DF

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3) Find, to the nearest tenth:

a) TU

b) TV

Solving Problems Using Trigonometric Ratios

Model Problem

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Exercise

3) Find KL. 4) Find GH and HJ.

Lesson Summary

1. Label the sides of the triangle, OPPOSITE, ADJACENT, and HYPOTENUSE.

2. Using SOH-CAH-TOA, determine which trig ratio to use in the problem.

3. Set up the trig ratio using the formula.

4. Cross-multiply and solve.

Exit Ticket Find JL.

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Homework

7) Find TV and TU.

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2)

3)

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Finding a Missing Angle When the Side Lengths are Known

Warm-Up

Finding Missing Angle Measures Using Inverse Functions

If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle.

Example #1

Find sin 35 °¿ Find sin−1( .573576¿)¿ _________________

Find cos18 °¿ Find cos−1(.951056¿)¿ _________________

Find tan22 °¿ Find sin−1( .404026¿)¿ _________________

What does the inverse function do? _____________________________________

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We say “sine inverse”

Example #2

Find the measure of angle A if:

1) sin A = 0.3456 2) tan A = 1.4552 3) cos A = 0.4995

Example #3

Find the measure of angle A to the nearest degree.

Guided Practice

Find the measure of angle P to the nearest degree.

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Practice Find each angle measure to the nearest degree.

1) Find m∠R. 2) Find m∠B.

3) Find m∠F. 4) Find m∠R.

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Solving Word Problems

Model Problem

Exercise

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Homework

13)

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