TRIG MINI-UNIT

BASIC TRIG RATIOS Trigonometric ratios are used with right triangles to solve for missing angles and/or sides.

The 3 main trig ratios are: Sine Cosine Tangent

LABELING TRIANGLE SIDES

Sides are labeled in reference to a designated angle, θ (“theta”)

Hypotenuse: the longest side. Always opposite the right angle.

Adjacent: side that is touching the angle θ.

Opposite: side across from the angle θ.

SOH CAH TOA

name rationotatio

nsine opp/hyp sin(θ)

cosine adj/hyp cos(θ)

tangent opp/adj tan(θ)

EVALUATING TRIG RATIOS Set up the ratio using the correct side lengths.

Reduce if possible. OR – divide and round your answer.

Depends on the directions given.

EXAMPLES: Find the value of each trig ratio.

sin A =

cos A =

tan A =

sin B =

cos B =

tan B =

EXAMPLES: Find the value of each trig ratio to the nearest ten-thousandth.

sin R =

cos R =

tan S =

INVERSE TRIG RATIOS

Inverse trig ratios are used to solve for missing angle measures.

They include: sin-1

cos-1

tan-1

On your calculator: hit “2nd” and then the trig button you need

EXAMPLES: When given a decimal value: Find each angle measure to the nearest degree.sinθ = 0.7193

cosθ = 0.3907

tanθ = 0.6009

EXAMPLES: When given a triangle: Set up the appropriate ratio, then use the inverse.

Find the measure of the indicated angle to the nearest degree.

Find the measure of the indicated angle to the nearest degree.

YOUR TURN! Find the measure of the indicated angle to the nearest degree.

1. tanθ = 1.6003

2.

SOLVING FOR MISSING SIDES

Set up trig ratio using the info given.

Solve for x. Example:

EXAMPLES: Find the missing side. Round to the nearest tenth.

YOUR TURN! Find the missing side. Round to the nearest tenth.

SPECIAL RIGHT TRIANGLES There are two types of “special right triangles.”

Their side lengths follow special rules.

45 – 45- 90 We can use the Pythagorean Theorem to verify the length of the hypotenuse if both legs are 1.

It follows that for any 45-45-90, the same relationships are true.

In general:

EXAMPLES: Find the missing side lengths. Leave answers in simplest radical form.

EXAMPLES:

4√2

YOUR TURN! Find the missing side lengths.

30-60-90 A 30-60-90 is half of an equilateral triangle.

That means the hypotenuse is twice the short leg.

We can use the Pythagorean Theorem to find the long leg.

It follows that for any 30-60-90, the same relationships are true.

In general:

60º

EXAMPLES: Find the missing side lengths.

Leave answers in simplest radical form.

EXAMPLES:

EXAMPLE:

3√2

x

YOUR TURN! Find the missing side lengths. Leave answers in simplest radical form.

MULTI-STEP TRIG PROBLEMS

Use trig to find the length of the common side.

Then, use trig again to solve for the designated side.

Example

x

57º 49º12

EXAMPLE:

12 x

66º

52º

YOUR TURN!

x

34º 59º

5

FINDING AREAS Remember: A = ½bh for triangles

Use trig to find the length of the base and height.

Then find the area. Example:

TRIG WORD PROBLEMS Hints for successful problem solving:

Draw a picture! Label all given information. Mark the angles or sides you need to

find. Use a different variable for each quantity.

Create a game plan! Solve using trig. Check that your answer is

reasonable. The hypotenuse is always the longest side!

COMMON VOCABULARY Angle of elevation – measured upward from the horizontal.

Angle of depression – measured downward from the horizontal.

BASIC EXAMPLE: A light house is 60 meters high with

its base at sea level. From the top of the lighthouse, the angle of depression of a boat is 15 degrees. A. How far is the boat from the foot of the light house?B. How far is the boat from the top of the lighthouse?

MULTI-STEP EXAMPLE: Katie and Sara are attending a

theater performance. From her seat, Katie looks down at an angle of 18 degrees to see the orchestra pit. Sara's seat is in the balcony directly above Katie. Sara looks down at an angle of 42 degrees to see the pit. The horizontal distance from Katie's seat to the pit is 46 ft. What is the vertical distance between Katie's seat and Sara's seat?