1 / 20

Right Triangle Trigonometry

Right Triangle Trigonometry. Section 13.1. Review right triangle trigonometry from Geometry and expand it to all the trigonometric functions Begin learning some of the Trigonometric identities. Today’s Objective. Evaluate trigonometric functions of acute angles.

karik
Download Presentation

Right Triangle Trigonometry

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Right Triangle Trigonometry Section 13.1

  2. Review right triangle trigonometry from Geometry and expand it to all the trigonometric functions Begin learning some of the Trigonometric identities Today’s Objective

  3. Evaluate trigonometric functions of acute angles. • Use fundamental trigonometric identities. • Use a calculator to evaluate trigonometricfunctions. • Use trigonometric functions to model and solvereal-life problems. What You Should Learn

  4. Trigonometry is based upon ratios of the sides of right triangles. The ratio of sides in triangles with the same angles is consistent. The size of the triangle does not matter because the triangles are similar (same shape different size). Right Triangle Trigonometry

  5. θ The six trigonometric functions of a right triangle, with an acute angle ,are defined by ratios of two sides of the triangle. hyp opp The sides of the right triangle are: adj  the side opposite the acute angle ,  the side adjacent to the acute angle ,  and the hypotenuse of the right triangle.

  6. Trigonometric Functions θ adj opp sin  = cos  = tan  = csc  = sec  = cot  = hyp adj hyp hyp adj opp adj opp hyp The trigonometric functions are opp adj sine, cosine, tangent, cotangent, secant, and cosecant. Note: sine and cosecant are reciprocals, cosine and secant are reciprocals, and tangent and cotangent are reciprocals.

  7. Another way to look at it… sin  = 1/csc csc = 1/sin cos = 1/sec sec = 1/cos tan = 1/cot cot = 1/tan Reciprocal Functions

  8. 5  12 Given 2 sides of a right triangle you should be able to find the value of all 6 trigonometric functions. Example:

  9. Example: Six Trig Ratios 5 4  3 sin  = sin α = cos  = cos α = tan  = cot  = cot α = tan α = sec  = csc  = sec α = csc α = Calculate the trigonometric functions for  . Calculate the trigonometric functions for . The six trig ratios are What is the relationship of α and θ? They are complementary (α = 90 – θ)

  10. Example: Using Trigonometric Identities hyp a Side a is opposite θ and also adjacent to 90○– θ . 90○– θ θ b sin  = and cos(90 ) = . So, sin  = cos (90 ). Note sin  = cos(90 ), for 0 <  < 90 Note that  and 90  are complementary angles. Note : These functions of the complements are called cofunctions.

  11. Consider an isosceles right triangle with two sides of length 1. 45 1 45 1 The Pythagorean Theorem implies that the hypotenuse is of length . Geometry of the 45-45-90 triangle Geometry of the 45-45-90 Triangle

  12. Example: Trig Functions for  30 2 1 30 sin 30 = = cos 30 = = tan 30 = = = cot 30 = = = adj opp hyp adj csc 30 = = = 2 sec 30 = = = hyp hyp adj opp adj opp Calculate the trigonometric functions for a 30 angle.

  13. Example: Trig Functions for  60 2 60○ 1 sin 60 = = cos 60 = = cot 60 = = = tan 60 = = = adj opp hyp adj sec 60 = = = 2 csc 60 = = = hyp hyp adj opp adj opp Calculate the trigonometric functions for a 60 angle.

  14. Some basic trig values

  15. Find the value of x for the right triangle shown Find Missing Side Length sin 60° = = 5 = x

  16. Solve ∆ABC Using a Calculator To Solve a/13 = tan 19 c/13 = sec 19 a= 4.48 and c = 13.8

  17. Applications Involving Right Triangles The angle you are given is the angle of elevation, which represents the angle from the horizontal upward to an object. For objects that lie below the horizontal, it is common to use the term angle of depression.

  18. A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.3. How tall is the Washington Monument? Using Trigonometry to Solve a Right Triangle Figure 4.33

  19. where x = 115 and y is the height of the monument. So, the height of the Washington Monument is y = x tan 78.3  115(4.82882)  555 feet. Solution

  20. An airplane flying at an altitude of 30,000 feet is headed toward an airport. To guide the airplane to safe landing, the airport’s landing system sends radar signals from the runway to the airplane at a 10 angle of elevation. How far is the airplane from the airport runway? 30,000 ft

More Related