4.3 Right Triangle Trigonometry

Right Triangle Trig Our second look at the trigonometric functions is from a ___________________ ___________________ perspective. The three sides of __________ triangle are the __________________, the __________________, and the ___________________. The __________________ side is across from the _______________ ________________. right triangle hypotenuse adjacent the right hypotenuse opposite side adjacent side hypotenuse right angle

Right Triangle Trig The __________________ side is opposite from the _______________ ________________. The __________________ side is next to the _______________ ________________. Using the lengths of these three sides, you can form the ______________ ratios that define the _____ _____________ ____________ of the acute angle. hypotenuse adjacent opposite given angle adjacent given angle six functions trigonometric six

Six Ratios Define the six trigonometric functions of the acute angle  To help us remember we use the acronym: SOH CAH TOA

Six Ratios NOTE: The functions in the second column are _______________________ of the _______________________ in the first column. Uh Oh… What do you think we should do if we only have two of the three sides of a right triangle? or Page 533 Problems 1-8, find the missing side reciprocalsfunctions Use Pythagorean Theorem!

Ex. 1 Evaluating Trigonometric Functions Find the values of the six trigonometric functions of , as shown in the figure 4 3

Ex. 2 Special Triangles: 45 – 45 – 90 Find the values of sin 45 , cos 45 , and tan 45 . 1 1

Ex. 3 Special Triangles: 30 – 60 – 90 Use the equilateral triangle shown to find the values of sin 60°, cos 60°, sin 30°, and cos 30°. Note a = 2 Page 533 Problems 1-20

SINES, COSINES, AND TANGENTS OF SPECIAL ANGLES

Co-functions Co-functions of complementary angles are… equal.

Ex 4Find the co-function of complementary angles. a.b.

Ex 4Find the co-function of complementary angles. c.d. Page 533 prob. 21 – 28

Ex 5 Evaluating Trigonometric Functions With a Calculator a.b. c. Page 533 prob

APPLICATIONS INVOLVING RIGHT TRIANGLES Angles of Elevation:  Means angles that are measured from bottom up. Angles of Depression:  Means from line of sight down.

Ex 7 A surveyor is standing 50 feet from the base of a large tree, as shown in the figure. The surveyor measures the angle of elevation to the top of the tree as 71.5 . How tall is the tree?

Ex. 8 A person is 200 yards from a river. Rather than walking directly to the river, the person walks 400 yards along a straight path to the river’s edge. Find the acute angle  between this path and the river’s edge, as illustrated. Page prob

Using a Calculator to Find an Angle Keys for TI-84: 1. Use the “2 nd ” key then sin to access the sin -1 option. 2. Use “2 nd ” then cos to access cos Use “2 nd ” then tan for tan -1. Keys for TI-Nspire 1. Press Trig button. 2. Select sin -1 option Ex 9 Find the angle that makes Page 534 problems

Ex. 10 A 12-meter flagpole casts a 9-meter shadow, as shown in the figure below. Find θ, the angle of elevation of the sun.