An Introduction to Trigonometry
An Introduction to Trigonometry
§13.1
Trig Identities
[pic] [pic] [pic]
[pic] [pic] [pic]
Example 1
Find the six trig functions for angle [pic].
Example 2
Find tanM when [pic]. Round to four decimal places.
Solving Right Triangles
SOH CAH TOA
Looking for a side of the triangle: sin, cos, tan
Looking for an angle of the triangle: [pic], [pic], [pic]
Example 3
a. [pic] b. [pic] c. [pic]
Example 4
Solve each triangle. Round each side to the nearest tenth and each angle to the nearest degree.
[pic]
Example 5
Solve each triangle. Round each side to the nearest tenth and each angle to the nearest degree.
[pic]
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 [pic]
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 [pic], how tall is APHS?
Pg 777, 1, 3, 7-41 odds
Angles and Angle Measure
§13.2
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
[pic]
Therefore, [pic]
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
Use [pic] for conversion factor
Example 4
Convert each degree measure to radian measure.
a. 60 b. 150
Convert Radians to Degrees
Use [pic] for conversion factor
Example 5
Convert each radian measure to degree measure.
a. [pic] b. [pic]
Example 6
Find one positive and one negative angles using radian measure that are coterminal to each.
a. [pic] b. [pic]
Pg 783,6-9,20-46 even
The Unit Circle
§13.2 (Extend)
However, the unit circle has r = 1. Therefore,
However, the unit circle has r = 1. Therefore,
Unit Circle Worksheet
Trigonometric Functions of General Angles
§13.3
Coordinate Plane
sin α = csc α =
cos α = 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 (2, 1).
(x, y)
(cos, sin)
Example 2
Find the exact values of each (notice multiples of 90, UNIT CIRCLE).
a. sin 90 b. cos 180 c. tan 90
d. sec 180 e. cot 270
Reference Angles
“All Students Take Calculus”
Example 3
Find the exact value using reference angles.
a. sin 150 b. cos 150
c. tan 240 d. [pic]
Pg 790, 5, 17-27, 32-42
The Law of Sines
§13.4
Oblique Triangle – a triangle without a right angle.
Law of Sines
[pic]
Example 1
Find the remaining parts of the triangle.
a. [pic] b. [pic]
Ambiguous Case (SSA or ASS)
[pic]
a < h a = h h < a < b [pic]
0 triangles 1 triangle 2 triangles 1 triangle
Example 3
Find the remaining parts of the triangle.
a. [pic]
b. [pic]
c. [pic]
Pg 276, 15-19,21-25 odds, 36(SSA)
The Law of Cosines
§13.5
Law of Cosines
[pic]
[pic]
[pic]
Problem with law of cosines (SAS)
1. Solve side first
2. Pick angle, check if realistic
3. If answer is not realistic, must do other angle first.
Example 1
Solve the remaining parts of the triangle.
a. [pic] b. [pic]
Pg 285, 13-25 odds,36
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