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Trigonometry

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Tara Morgen

on 4 June 2015

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Transcript of Trigonometry

Period= 2 pi
Scale= pi/2
Amplitude= 1
Zeros= x=pi/2+k, where k is an integer
Asymptotes= none
Domain= all real numbers
Range= -1 is less than or equal to y is less than or equal to 1
Trigonometry

What Is Trigonometry?
The literal definition of trigonometry is "the measuring (of angles and sides) of triangles."

A more technical definition is "the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles."
Applications
In the following example, x represents the height of the tree. The height of the tree is unknown. The tree's shadow measures 30 meters. The angle of elevation is 50°. A right angle is formed between the ground and the trunk of the tree.

All triangles have combined angle measures of 180°. This problem has provided us with 2 angle measures already: 50° & 90°.

50° + 90° = 140°
180° - 140° = 40°

Through simple addition and subtraction, we've discovered that the angle of inclination is 40°. Now to solve for the sides.

cos(50°) = x/30
30cos(50°) = x
30(0.64278760968) = x
19.2836282906 = x
19 = x

If we were asked to complete the triangle, we would do using the Pythagorean Theorem.

a2 + b2 = c2
(19)2 + (30)2 = c2
361 + 900 = c2
1261 = c2
35.5105618091 = c
36 = c
Trigonometric Functions
SOH CAH TOA
Special Right Triangles
30 - 60 - 90 Triangle
45 - 45 - 90 Triangle
SOH CAH TOA is a mnemonic used to help remember how to compute the sine, cosine, and tangent of an angle.

- SOH stands for sine is equal to opposite over hypotenuse.
- CAH stands for cosine is equal to adjacent over hypotenuse.
- TOA stands for tangent is equal to opposite over adjacent.
For this triangle, solving for sin A, cos A, and tan A would require using SOH CAH TOA.
sin A= opp/hyp = a/c
cos A= adj/hyp = b/c
tan A= opp/adj = a/b

The sides of this triangle are always in the following ratio:
x : x radical 3 : 2x

The sides of this triangle are always in the following ratio:
x : x: x radical 2
* It is an isosceles triangle, so two angles are congruent and two corresponding sides are also congruent.
Radians and Degrees
The Unit Circle
Angles can also be measured in units called radians
There are 2 pi radians in one complete revolution
Unit of
measurement
for angles
Angles can be measured in units called degrees
There are 360 degrees in one complete revolution
1 rad= (180/pi)°degrees equals approximately 57.3
Ex:
Ex:
90 degrees = pi/2 rad
( 3 pi ) / 2 = 270 degrees
Centered at (0,0) with a radius of 1
Equation for the unit circle is: x^2 + y^2 = 1
We can use the unit circle in order to evaluate trigonometric functions
Inverse Trigonometric Functions
Solving Trigonometric Equations
Basic Trigonometric Identities
Graphing Trigonometric Functions
Pythagorean identities
In order to solve a trigonometric equation, you will need to know how to solve basic algebraic two-step equations.
You will also need to know how to use reference angles along with ASTC.
y = sin (x)
y = cos (x)
y = tan (x)
y = csc (x)
y = sec (x)
y = cot (x)
Period= 2 pi
Scale= pi/2
Amplitude= 1
Zeros= x=kpi, where k is an integer
Asymptotes= none
Domain= all real numbers
Range= -1 is less than or equal to y is less than or equal to 1
Period= pi
Scale= pi/4
Amplitude= none
Zeros= x+kpi, where k is an integer
Asymptotes= x=pi/2+kpi, where k is an integer
Domain= all real numbers except x=pi/2+kpi, where k is an integer
Range= all real numbers
Period= 2pi
Scale= pi/2
Amplitude= none
Zeros= none
Asymptotes= x=kpi, where k is an integer
Domain= all real numbers except x=kpi, where k is an integer
Range= y is greater than or equal to 1 or y is less than or equal to -1
Period= 2pi
Scale= pi/2
Amplitude= none
Zeros= none
Asymptotes= x=pi/2+kpi, where k is an integer
Domain= all real numbers except x=pi/2+kpi, where k is an integer
Range= all real numbers
Period= pi
Scale= pi/4
Amplitude= none
Zeros= x=pi/2+kpi, where k is an integer
Asymptotes= x=kpi, where k is an integer
Domain= all real numbers except x=kpi, where k is an integer
Range= all real numbers
Reciprocal identities
Ratio identities
Methods to Use for Solving Trigonometric Equations
Example Problem: Solve cos2x + 6 sinx - 3=0

Taking the Square Root on Both Sides
This method will leave you with four answers (2 positive and 2 negative) since there is a square involved, if between 0 degrees and 360 degrees.
Solving Quadratic Functions in Trigonometric Equations
This can be done by factoring, completing the square, quadratic formula, graphing, and looking at the square roots.
Substituting an Angle
When you are not given an angle by itself, you have to substitute a letter, theta (x) for the value.

Ex: Let theta (x) = A + 40 degrees

Solving by Factoring
Some equations can only be solved by factoring or using the distributive property to pull out a specific term to one side so you can then solve.
Approximating Solutions to Trigonometric Equations
Often, you will not find exact values when solving. Instead, we have to use the inverse of the function and then round it to get the resulting reference angle that will be used to solve the equation.
Step One
Step Two
Use the quadratic formula to get sinx
x=(-6±square root(6^2-4(-2)(2))/2(-2)
sinx= (3±square root 5)/ (2)
Step Three
sinx= (3+⁡ 5)/2 = 2.618033989 (out of range of sine)

sinx= (3- 5)/2 = .3819660113
Step Four
sin^(-1) .3819660113= 22.5 degrees

The positive value results in two answers for x, where sine is positive, quadrants I and II.

x= 22.5 degrees
x= 157.5 degrees


A trigonometric identity is a formula that uses trigonometric functions and is true no matter which values are chosen. Below are basic identities.
NOTE: DO NOT TAKE THE INVERSE OF A NEGATIVE EXACT VALUE
These graphs represent trigonometric functions and their inverses.
Use the Reference Sheet to substitute cos 2x with 1-2sin^2x

1-2sin^2x+6sinx-3=0
-2sin^2x+6sinx-2=0
Elijah Fritz / Hannah Smith / Jessica Goodale / Tara Morgen
Summary of
The six trigonometric functions are defined as follows:

sin(x) = y/r csc(x) = 1/sin(x) = r/y

cos(x) = x/r sec(x) = 1/cos(x) = r/x

tan(x) = y/x cot(x) = 1/tan(x) = x/y
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