Warm Up Find the missing sides of the following triangles:

Answers Find the missing sides of the following triangles:

Trigonometry Trigonometry was developed by Greek mathematicians over 2000 years ago. It was created to study astronomy. By understanding the relationships between sides and angles, astronomers could map the movements of planets and stars.

First, we will focus on right triangles! Today, you will learn how to: solve for a missing side of a right triangle solve for a missing angle of a right triangle Later in the unit, we will work with non-right triangles.

Labeling Triangles for Trig

You try…

Tangent of ∠ A A Abbreviated as…

Sine of ∠ A Abbreviated as…

Cosine of ∠ A Abbreviated as… adjacent

We now have three useful trig ratios:

soh cah toa

SOH-CAH-TOA Ex 1: Write the 3 Trig Functions for each angle. (A and C) We never use the 90 o angle! sin A = sin C = cos A = cos C = tan A = tan C =

Example 2: Finding a Side Length Use a Trig Function and solve using Algebra! Find x. When you need to use your calculator, ALWAYS make sure it’s in DEGREE mode

Use Trig Function and solve using Algebra! Find x. Example 3: Finding a Side Length

How do we find the measure of an angle? So far, we have two ways: – Protractor – Angle Sum Theorem If you know two angles, you can find the third by subtracting known angles from 180 But what if we only know two sides of the triangle?

Only need two sides! Find the measure of angle C – What sides do we know? – Which trig function goes with those sides?

Now our three trig functions can be used to find the measure of an angle!

Ex. 6 Find Angle M

Am I finding an angle or a side?Angle Sine, cosine or tangent? Use the inverse function. SideSine, cosine, or tangent? Use algebra to solve for the missing side.

Example 1: Hot Air Balloon As a hot-air balloon began to rise, the ground crew drove 1.2 mi to an observation station. The initial observation from the station estimated the angle between the ground and the line of sight to the balloon to be 30º. Approximately how high was the balloon at that point? ‘

The balloon is approximately 0.7 mi, or 3696 ft, high.

Example 2: Width of a River A surveyor can measure the width of a river by standing on point C and taking a sighting at point A on the other side. After turning 90° and walking 200m, he takes another sighting from point B. Angle B is measured and found to be 90°. What is the width of the river? The width of the river is approximately 73 meters.

Example 3: Painting a House For safety reasons, the base of a 25 ft must be 6.5 ft from the base of wall. At what angle with the ground should a painter place his ladder in order to maximize his height? Thus when the ladder is in its safest position, it makes an angle of about 75º with the ground.