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Math 125 - Final Exam

The document contains instructions and formulas for solving problems on trigonometric functions and identities for a final exam. It includes definitions of trig functions, inverse trig functions, trig identities, solving right triangles using definitions and laws of sines, guidelines for sketching graphs of trig functions including sine, cosine, secant and cosecant functions. It also provides formulas and properties of trig functions, inverse trig functions, double-angle formulas, sum and difference formulas.

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Phuc Dang
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171 views10 pages

Math 125 - Final Exam

The document contains instructions and formulas for solving problems on trigonometric functions and identities for a final exam. It includes definitions of trig functions, inverse trig functions, trig identities, solving right triangles using definitions and laws of sines, guidelines for sketching graphs of trig functions including sine, cosine, secant and cosecant functions. It also provides formulas and properties of trig functions, inverse trig functions, double-angle formulas, sum and difference formulas.

Uploaded by

Phuc Dang
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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Final Exam Name:_________________________________________ Show all your work to receive full credit 1) (20 points) What are the

Fundamental Identities? (7.5 points)

Fill in the blank:

(12.5 points)

2)

(5 points) Convert to degrees Hint: Use

3)

(10 points) Given a point ( Hint: Calculate radius ) Find the exact values of six trigonometric functions

4)

(5 points) Suppose Which quadrant does

lie in?

5)

(10 points) Establish the identity:

6)

(10 points) Given a point Hint: Calculate radius Find the exact values of six trigonometric functions

7)

(20 points) Graph

8)

(10 points) Establish the identity:

9) (10 points) Find the exact value of:

)+

10)

(10 points) Establish the identity: = cos

11)

(20 points)

Given

where where

Find the exact value of: a) sin( b)

12)

(10 points) Establish the identity: =

13)

(15 points) a) Given b = 4, B = 10o Find a, c, and A by using Definition of Right Triangle (7.5 points)

b)

Given B = 20o, C = 70o, a = 1 Solve the triangle using Law of Sine

(7.5 points)

USING THESE FORMULAS WHEN r = 1:

sint = y cost = x
USING THESE FORMULAS WHEN r :

tant = cott =

sect = csct =

x2 + y2 = r2 sin = cos =
DEFINITION OF RIGHT TRIANGLE

tan = cot =

sec = csc =

sin = cos =

tan = cot =

sec = csc =

LAW OF SINES For any triangle ABC with sides a, b ,c

or

with A + B + C = 180o
INVERSE TRIGONOMETRIC FUNCTIONS y = sin-1x y = cos-1x means means siny = x cosy = x , where , where 0 and and 1 1

PROPERTITIES OF INVERSE TRIGONOMETRIC FUNCTIONS

f-1(f(x)) = sin-1(sinx) = x f(f-1(x)) = sin(sin-1x) = x f-1(f(x)) = cos-1(cosx) = x f(f-1(x)) = cos(cos-1x) = x

where where where where 1 1

3a 3b 3c 3d

DOUBLE ANGLE FORMULAS cos


HALF ANGLE FORMULAS

=
tan = or tan =

SUM AND DIFFERENCE FORMULAS


( ( SUM TO PRODUCT FORMULAS c c

Guidelines for Sketching Graphs of Sine and Cosine Functions To graph y = A sin( x) or y = A cos( x), with > 0, follow these steps: Step 1: Use the amplitude to determine that maximum and minimum values of the function. Step 2: Find the period, . Then divide the interval [0, ] into four equal subintervals. a) Find the midpoint of the interval by adding the x-values of the endpoints and dividing by 2. b) Find the midpoints of the two subintervals found in a), using the same procedure. Step 3: Make the table of these subintervals to obtain five key points on the graph. Step 4: Connect these points with a sinusoidal graph to obtain the graph of one circle and extend the graph in each direction to make it complete. Guidelines for Sketching Graphs of Secant and Cosecant Functions To graph y = A sec( x) or y = A csc( x), with > 0, follow these steps: Step 1: Graph the corresponding reciprocal function as a guide, using dashed curve To graph Use as a guide y = A sec( x) y = A cos( x) y = A csc( x) y = A sin( x) Step 2: Sketch the vertical asymptotes. They will have equations of the form x = k, where k in an x-intercept of the graph of the guide function. Step 3: Sketch the graph of the desired function by drawing the typical U-shaped branches between the adjacent asymptotes. The branches will be above the graph of the guide function when the guide function values are positive and below the graph of the guide function when the guide function values are negative.

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