Warm Up Copyright © 2009 Pearson Education, Inc.

Trigonometry Trigonometry was developed by Greek mathematicians over 2000 years ago. It was created to study astronomy. Trigonometry is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides.

How are we going to use trigonometry? Our goals for the next lesson: solve for a missing side of a right triangle solve for a missing angle of a right triangle

Labeling Triangles for Trig

You try…

Sine of ∠ A Abbreviated as…

Cosine of ∠ A Abbreviated as… adjacent

Tangent of ∠ A A Abbreviated as…

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

Ask Yourself Am I finding an angle or a side?Angle: Use Inverse TrigCan I use Sin, Cos, Tan? Find the inverse of your fraction Side: Use reg. TrigCan I use Sin, Cos, Tan? Use Algebra to solve for 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? ‘ Draw the situation, label the acute angle and length of the adjacent side.

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

Example 2: Width of a River

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 3: Inclination

A paint crew has purchased new 30-ft extension ladders. The manufacturer states that the safest placement on a wall is to extend the ladder to 25 ft and to position the base 6.5 ft from the wall. What angle does the ladder make with the ground in this position? Example 4: Ladder Position

Copyright © 2009 Pearson Education, Inc. Thus when the ladder is in its safest position, it makes an angle of about 75º with the ground. Use a calculator to find the acute angle whose cosine is 0.26: