Right Triangle Trigonometry!

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 First, put stickman next to angle, then you always have hyp across from 90, and adj, opp

You try… Have students come to board and label sides. REMIND STUDENTS OF RATIOS. ONE THING OVER ANOTHER. BOYS TO GIRLS, ETC.

Tangent of ∠A Abbreviated as… A Define the ratio “opp/adj” as “Tangent” ∠ A

Sine of ∠A Abbreviated as… Define opp/hyp as sine

Cosine of ∠A Abbreviated as… Define adj/hyp as cosine adjacent

We now have three useful trig ratios:

soh cah toa

SOH-CAH-TOA cos A = cos C = tan A = tan C = Ex 1: Write the 3 Trig Functions for each angle. (A and C) We never use the 90o angle! sin A = sin C = cos A = cos C = tan A = tan C = SIN a, COS a, tan a……sin c, cos c, 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 Draw stick figure, what sides do I know? Hyp and opp. So use soh=sin. Sin35=opp/hyp=x/20. now use algebra. CROSS MULTIPLY!!! 20*SIN35=x*1. use calculator in degree mode

Example 3: Finding a Side Length Use Trig Function and solve using Algebra! Find x. Stick man. Sides? Adj and hyp. Use cah=cos. Cos72/1=adj/hyp=6/x. cross multiply. xcos73=6. x=6/cos72

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 Trig Can I use Sin, Cos, Tan? Find the inverse of your fraction Side: Use reg. Trig 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 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Example 4: Ladder Position 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?

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