Right Triangle Trigonometry

1. Right Triangle Trigonometry

2. Hypotenuse Side opposite  Side adjacent to  The sides of a right triangle Take a look at the right triangle, with an acute angle, , in the figure below. Notice how the three sides are labeled in reference to . In this section, we will be studying special ratios of the sides of a right triangle, with respect to angle, .

3. These ratios are better known as oursix basic trig functions:

4. Definitions of the Six Trigonometric Functions

5. Definitions of the Six Trigonometric Functions To remember the definitions of sine, cosine and tangent, we use the acronym : “SOH CAH TOA”

6.  5 9 Example Find the exact value of the six trig functions of  in the triangle below: First find the length of the hypotenuse using the Pythagorean Theorem.

7. Example Given that  is an acute angle and , find the exact value of the five remaining trig functions of .

8. Example Find the value of sin  given cot  = 0.387, where  is an acute angle. (Divide ratio and give answer to three significant digits.)

9. 45º 1 45º 1 Special Right Triangles The 45º- 45º- 90º Triangle Ratio of the sides: Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 45 sin 45 = ≈ csc 45 = ≈ cos 45= ≈ sec 45 = ≈ tan 45 = cot 45 =

10. 30º 2 60º 1 Special Right Triangles The 30º- 60º- 90º Triangle Ratio of the sides: Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 30 sin 30= csc 30 = cos 30 = ≈ sec 30 = ≈ tan 30 = ≈ cot 30 = ≈

11. 30º 2 60º 1 Special Right Triangles The 30º- 60º- 90º Triangle Ratio of the sides: Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 60 sin 60 = ≈ csc 60 = ≈ cos 60 = sec 60 = tan 60 = ≈ cot 60 = ≈

13. Using the Calculator to Evaluate Trig Functions To evaluate trig functions of acute anglesother than 30, 45, and 60, you will use the calculator. Your calculator has keys marked , , and . **Make sure the MODE is set to the correct unit of angle measure. (Degree vs. Radian) Example: Findto two decimal places. 13

14. Angles and Accuracy of Trigonometric Functions

15. Using the Calculator to Evaluate Trig Functions To find the values of the remaining three functions (cosecant, secant, and tangent), use the reciprocal identities. For reciprocal functions, you may use the  button, but DO NOT USE THE INVERSE FUNCTIONS (e.g. SIN-1 )! Example: 1. Find2. Find (to 3 significant dig) (to 4 significant dig) X

16. The Inverse Trigonometric Functions The inverse trig functions give the measure of the angle if we know the value of the function. Notation:The inverse sine function is denoted as sin-1x or arcsin x. It means “the angle whose sine is x”. The inverse cosine function is denoted as cos-1x or arccos x. It means “the angle whose cosine is x”. The inverse tangent function is denoted as tan-1x or arctan x. It means “the angle whose tangent is x”.

17. Example of Inverse Trig Function • For example, will yield the acute angle whose sine is . • You can think of this as the related equation

18. Examples using common angles • Evaluate the following inverse trig functions using the special triangles (you do not need a calculator): • 1) 2)

19. Examples using the calculator Evaluate the following inverse trig functions using the calculator. Give answer in degrees. Round appropriately.

20. Examples using the calculator Evaluate the following inverse trig functions using the calculator. Give answer in degrees. Round appropriately.

21. Since you are looking for the side adjacent to 52º and are given the hypotenuse, you could use the _____________ function. 52º 9.6 y Using Trig Ratios to Find Missing Parts of Right Triangles Example: Solve for y in the right triangle below: Solution:

22. Solving Right Triangles • To solve a right triangle is to find any missing angles and any missing sides. • You will always be given 3 parts, and you will need to find 3 parts. • The angles are labeled using capital letters A, B, & C. Use angle C to represent the right angle. Angles A and B represent the acute angles. • The sides are labeled using lowercase letters a, b, & c. Each side is labeled with respect to its opposite angle.

23. A= 40.7° b c B C a=8.20” Example Solve the right triangle with the indicated measures. Solution Answers:

24. A c=35.4 b C a=25.8 B Example Answers:

25. 36° 36° 8.6 m Example 3. Find the altitude of the isosceles triangle below.

26. Example 4. Solve the right triangle with Answers:

27. Angle of Elevation and Angle of Depression The angle of elevation for a point above a horizontal line is the angle formed by the horizontal line and the line of sight of the observer at that point. The angle of depression for a point below a horizontal line is the angle formed by the horizontal line and the line of sight of the observer at that point. Horizontal line Angle of depression Angle of elevation Horizontal line

28. 108 m h 42.3° Example A guy wire of length 108 meters runs from the top of an antenna to the ground. If the angle of elevation of the top of the antenna, sighting along the guy wire, is 42.3° then what is the height of the antenna? Give answer to three significant digits. Solution