13.1Introduction to Trigonometry Algebra 2 Goal 1: Find values of trigonometric functions for acute angles. Goal 2: Solve problems involving right triangles. B
c a
θ A
C
b
Trigonometric functions are ratios of the sides of a right triangle. There are six trig ratios for every acute angle in a right triangle.
sine =
sin _____
cosecant = csc _____
cosine = cos _____
secant =
sec _____
tangent = tan _____
cotangent = cot _____
SOH-CAH-TOA sine =
cosine =
tangent =
Example 1 Find the six trig functions for angle . Round to four decimal places. 3 5 3
6
Example 2 Find tanM when cos M
10 . Round to four decimal places. 14
Solving Right Triangles SOH CAH TOA Looking for a side of the triangle: sin, cos, tan Looking for an angle of the triangle: sin 1 , cos1 ,
tan1
Example 3 Solve. x a. cos56 7
b. sin x
7.6 12.1
c. tan 32
15.9 x
Example 4 Solve each triangle. Round each side to the nearest tenth and each angle to the nearest degree.
a = 5, b = 4
Example 5 Solve each triangle. Round each side to the nearest tenth and each angle to the nearest degree. A = 30, b = 8
Angle of Elevation – the angle between the line of sight and the horizontal when the observer looks upward.
Example 6 At the circus, a person in the audience at ground level watches the high wire routine. A 5’6” tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member’s line of sight to the top of the acrobats head is
27
Angle of Depression – the angle between the line of sight when an observer looks downward.
Example 7 Mark was lying down on top of APHS looking down at his Algebra 2 book which he left on the ground. If the book is 24 feet away from the base of the school with an angle of depression of 57 , how tall is APHS?
13.2 Angles and Angle Measure Algebra 2 Goal 1: Change radian measure to degree measure and vice versa Goal 2: Identify coterminal angles Initial Side – a ray fixed along the positive x-axis
Terminal Side – a ray that can rotate about the origin.
Degree measure of an angle – is the number of degrees in the intercepted arc of a circle centered at the vertex. The degree measure is positive if the rotation is counterclockwise and negative if the rotation is clockwise.
Coterminal Angles – angles α and β are coterminal if they have the same terminal side. *coterminal angles differ by a multiple of 360* Example 1 Find the degree measures of one positive and one negative angle that are coterminal with each given angle. a. 50º
b. -120 º
Example 2 Determine whether the given pair of angles is coterminal. a. 190 º, -170 º
b. 150 º, 880 º
Example 3 Name the quadrant in which the angles lies. a. 740 º
b. -510 º
Unit Circle r = 1 C 2r Therefore, C 2 s
α = angle in degrees s = radian measure of α s = α on the unit circle
Radian Measure – of the angle α in standard position is the directed length of the intercepted arc on the unit circle.
Convert Degrees to Radians
Convert Radians to Degrees
Use for conversion factor 180
Use 180 for conversion factor
Example 4 Convert each degree measure to radian measure. measure.
Example 5 Convert each radian measure to degree
a. 60
a. 2 3
b. 150
Example 6 Find one positive and one negative angles using radian measure that are coterminal to each. a.
4
b. 5 6
b. 5 4
13.2 The Unit Circle – Part 2
Goal: Determine the standard angles of the unit circle Unit Circle – 45º
However, the unit circle has r = 1. Therefore, 2
1
45 45
1
Unit Circle – 30º
However, the unit circle has r = 1. Therefore,
2 1 30
30 3
13.3 Trigonometric Functions of General Angles Algebra 2
Goal 1: Find values of trigonometric functions for general angles Goal 2: Use trigonometric identities to find values of trigonometric functions
Coordinate Plane sin α =
csc α =
(x, y) cos α =
r
sec α =
tan α =
cot α =
Example 1 Find the values of the six trigonometric functions of the angle α in standard position whose terminal side passes through A. (2, 1). B. ( - 3, 4)
(x, y) (cos, sin)
Example 2 Find the exact values of each (notice multiples of 90, UNIT CIRCLE). a. sin 90º
b. cos 180º
c. tan 90º
d. sec 180º
e. cot 270º
f. csc 0º
Reference Angles
sin + cos tan -
sin + cos + tan +
sin cos tan +
sin cos + tan -
“All Students Take Calculus”
Example 3 Find the exact value using reference angles. a. sin 150º
b. cos -150º
c. tan 240º
4 d. sin 3
e. sin 120º
5 f. cos 4
g. sin 6
11 h. tan 6
Pg 790, 5, 17-27, 32-42