Unit 8 – Right Triangle Trig Trigonometric Ratios in Right Triangles

Trigonometric Ratios are based on the Concept of Similar Triangles!

All 45º- 45º- 90º Triangles are Similar! 45 º 1 45 º 2 45 º 1

All 30º- 60º- 90º Triangles are Similar! 2 60º 30º 1 60º 30º 4 60º 30º 2 1 ½

Trigonometric functions -- the ratios of sides of a right triangle.  Similar Triangles Always Have the Same Trig Ratio Answers! adjacent opposite hypotenuse c b leg SINE COSINE  TANGENT leg a They are abbreviated using their first 3 letters

Oh, I'm acute! This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle.  5 4 So am I!  3

INE OSINE ANGENT PPOSITE DJACENT YPOTENUSE PPOSITE DJACENT YPOTENUSE Here is a mnemonic to help you memorize the ratios.  c b opposite SOHCAHTOA SOHCAHTOA  adjacent INE a OSINE ANGENT PPOSITE DJACENT YPOTENUSE PPOSITE DJACENT YPOTENUSE

Let's try finding some trig functions with some numbers. It is important to note WHICH angle you are talking about when you find the value of the trig function.  Let's try finding some trig functions with some numbers. hypotenuse 5 c b 4 opposite opposite  adjacent a 3 sin  = Use a mnemonic and figure out which sides of the triangle you need for tangent. Use a mnemonic and figure out which sides of the triangle you need for sine. tan  =

How do the trig answers for  and  relate to each other? hypotenuse c 5 b 4 opposite opposite  adjacent a 3

10.8 9 A 6 Find the sine, the cosine, and the tangent of angle A. Give a fraction and decimal answer (round to 4 places). 10.8 9 A 6 Now, figure out your ratios.

Tangent 14º 0.25 The Tangent of an angle is the ratio of the opposite side of a triangle to its adjacent side. hypotenuse 1.9 cm opposite adjacent 14º 7.7 cm Tangent 14º 0.25

Tangent A = 3.2 cm 24º 7.2 cm Tangent 24º 0.45

As an acute angle of a triangle approaches 90º, its tangent Tangent A = As an acute angle of a triangle approaches 90º, its tangent becomes infinitely large very large Tan 89.9º = 573 Tan 89.99º = 5,730 etc. very small

Sine A = Cosine A = Since the sine and cosine functions always have the hypotenuse as the denominator, and since the hypotenuse is the longest side, these two functions will always be less than 1. Sine A = Cosine A = Sine 89º = .9998 A Sine 89.9º = .999998

Sin α = 7.9 cm 3.2 cm 24º Sin 24º 0.41

Cosine β = 7.9 cm 46º 5.5 cm Cos 46º 0.70

Ex. Solve for a missing value using a trig function. tan 20 55 ) Now, figure out which trig ratio you have and set up the problem.

Ex: 2 Find the missing side. Round to the nearest tenth. 80 ft x tan 80  ( 72 ) ) = Now, figure out which trig ratio you have and set up the problem.

Ex: 3 Find the missing side. Round to the nearest tenth. Now, figure out which trig ratio you have and set up the problem.

Ex: 4 Find the missing side. Round to the nearest tenth. 20 ft x

Finding an angle. (Figuring out which ratio to use and getting to use the 2nd button and one of the trig buttons.)

Ex. 1: Find . Round to four decimal places. tan 17.2 9 ) 17.2 9 Now, figure out which trig ratio you have and set up the problem. Make sure you are in degree mode (not radians).

Ex. 2: Find . Round to three decimal places. 7 2nd cos 7 23 ) 23 Make sure you are in degree mode (not radians).

Ex. 3: Find . Round to three decimal places. 200 400 2nd sin 200 400 ) Make sure you are in degree mode (not radians).

When we are trying to find a side we use sin, cos, or tan. When we are trying to find an angle we use sin-1, cos-1, or tan-1.

4 x A plane takes off from an airport an an angle of 18º and a speed of 240 mph. Continuing at this speed and angle, what is the altitude of the plane after 1 minute? After 60 sec., at 240 mph, the plane has traveled 4 miles 4 x 18º

SohCahToa Soh Sine A = Sine 18 = 0.3090 = x = 1.236 miles or 0.3090 = 1 x = 1.236 miles or 6,526 feet 4 x opposite hypotenuse 18º

An explorer is standing 14.3 miles from the base of Mount Everest below its highest peak. His angle of elevation to the peak is 21º. What is the number of feet from the base of Mount Everest to its peak? Tan 21 = 0.3839 = 1 x = 5.49 miles = 29,000 feet x 21º 14.3

A swimmer sees the top of a lighthouse on the edge of shore at an 18º angle. The lighthouse is 150 feet high. What is the number of feet from the swimmer to the shore? 0.3249x = 150 Tan 18 = 0.3249 0.3249 0.3249 = 1 X = 461.7 ft 150 18º x

A dragon sits atop a castle 60 feet high. An archer stands 120 feet from the point on the ground directly below the dragon. At what angle does the archer need to aim his arrow to slay the dragon? Tan x = Tan x = 0.5 Tan-1(0.5) = 26.6º 60 x 120

Solving a Problem with the Tangent Ratio We know the angle and the side adjacent to 60º. We want to know the opposite side. Use the tangent ratio: h = ? 1 2 60º 53 ft Why?

Ex. A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree? tan 71.5° ? tan 71.5° 71.5° y = 50 (tan 71.5°) 50 y = 50 (2.98868)

Ex. 5 A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge? cos 60° x (cos 60°) = 200 200 60° x x X = 400 yards

Trigonometric Functions on a Rectangular Coordinate System x y q Pick a point on the terminal ray and drop a perpendicular to the x-axis. r y x The adjacent side is x The opposite side is y The hypotenuse is labeled r This is called a REFERENCE TRIANGLE.

Trigonometric Ratios may be found by: 45 º 1 Using ratios of special triangles For angles other than 45º, 30º, 60º you will need to use a calculator. (Set it in Degree Mode for now.)