A)Describe how to use this formula: theta=a/r to determine the MEASURE, in radia

Question

A)Describe how to use this formula:

theta=a/r to determine the MEASURE, in radians, of an angle of one rotation.

b)Explain your steps to CONVERT 150 degrees to RADIANS as fraction of pi.

2)FInd the EXACT RATIO for the following TRIGONOMETRIC expressions USING a DIAGRAM.

a)  sec  -4pi/3

b)  csc  5pi/4

3)If a circle has a radius of 65 m, determine the ARC length for a CENTRAL angle of 1.25 radians.

Simplify

4a) sin x/ tanx cos x

b)sin2x/1-cos2x

c) 4sin x/2 cosx/2

5)SHOW HOW YOU CAN UYSE THE COMPOUND AND DOUBLE ANGLE FORMULAS TO FIND AN EXPRESSION FOR EITHER sin3theta or cos3theta?

6)Prove the identities:

a)  tanx  + 1/tanx=1/sinxcosx

b)  tantheta-1=  sin^2theta-cos^2theta/sin theta costheta+ cos^2theta

7)If costheta= -2/root7, 0<theta<pi, FIND exact value for:

a) cos2theta

b)sin theta/2

c) sin(theta + pi/4)

Explanation / Answer

Verify the following identity: tan(x)+1/(tan(x)) = 1/(sin(x) cos(x)) Put the fractions in 1/(tan(x))+tan(x) over a common denominator. Put 1/(tan(x))+tan(x) over the common denominator tan(x): 1/(tan(x))+tan(x) = (1+tan(x)^2)/(tan(x)): (1+tan(x)^2)/(tan(x)) = ^?1/(cos(x) sin(x)) Eliminate the denominators on both sides. Cross multiply: cos(x) sin(x) (1+tan(x)^2) = ^?tan(x) Express both sides in terms of sine and cosine. Write tangent as sine/cosine: cos(x) sin(x) (1+ (sin(x))/(cos(x)) ^2 ) = ^?(sin(x))/(cos(x)) Simplify the left hand side. cos(x) sin(x) (1+((sin(x))/(cos(x)))^2) = cos(x) sin(x) (1+sin(x)^2/cos(x)^2): cos(x) sin(x) (1+sin(x)^2/cos(x)^2) = ^?(sin(x))/(cos(x)) Put the fractions in 1+sin(x)^2/cos(x)^2 over a common denominator. Put 1+sin(x)^2/cos(x)^2 over the common denominator cos(x)^2: 1+sin(x)^2/cos(x)^2 = (cos(x)^2+sin(x)^2)/cos(x)^2: cos(x) sin(x) (cos(x)^2+sin(x)^2)/cos(x)^2 = ^?(sin(x))/(cos(x)) Cancel down (cos(x) sin(x) (cos(x)^2+sin(x)^2))/cos(x)^2. Cancel cos(x) from the numerator and denominator. (cos(x) sin(x) (cos(x)^2+sin(x)^2))/cos(x)^2 = (cos(x) sin(x) (cos(x)^2+sin(x)^2))/(cos(x) cos(x)) = (sin(x) (cos(x)^2+sin(x)^2))/(cos(x)): (sin(x) (cos(x)^2+sin(x)^2))/(cos(x)) = ^?(sin(x))/(cos(x)) Eliminate the denominators on both sides. Multiply both sides by cos(x): sin(x) (cos(x)^2+sin(x)^2) = ^?sin(x) Cancel common terms. Divide both sides by sin(x): cos(x)^2+sin(x)^2 = ^?1 Use the Pythagorean identity on cos(x)^2+sin(x)^2. Substitute cos(x)^2+sin(x)^2 = 1: 1 = ^?1 Come to a conclusion. The left hand side and right hand side are identical: Answer: | | (identity has been verified)

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