Comprehensive Solutions: Trigonometry, Functions, Identities

This assignment provides detailed solutions to various trigonometry and function problems. It includes finding the length of an arc and the area of a sector, simplifying trigonometric expressions using identities, verifying trigonometric identities by transforming one side of the equation to match the other, and using fundamental identities to find values of trigonometric functions under given conditions. The assignment also covers finding exact values of x that satisfy trigonometric equations, determining amplitude, period, and phase shift for trigonometric functions, sketching graphs of trigonometric functions including asymptotes for tangent functions, and finding exact values of remaining parts of a right triangle using trigonometric relationships. Solved examples include problems related to csc, cos, cot, sin and tan trigonometric functions. This resource is perfect for students seeking assistance with their trigonometry assignments; find more solved examples and past papers on Desklib.

Surname 1
Name
Instructor’s Name
Course Code
Date
Assignment 11
Exercise 31-32: (a) Find the length of the arc of the colored sector in the figure. (b) Find the area
of the sector.
Solution
(a) Length of an arc = θ
360 ×2 πr
120
360 ×2 π × 9=18.850 cm
(b) Area of sector = θ
360 × πr2
120
360 × π ×92=84.823 cm2
Exercise 39-44: Simply the expression.
cot2 α−4
cot2 α−cot α−6
= ( cot α+2 ) ( cot α −2 )
(cot α +2) ( cot α−3 )
= cot α −2
cot α −3
Exercise 51-74: Verify the identity by transforming the left-hand side into the right-hand side.
58. cos2 2θ−sin2 2θ=2 cos2 2θ−1
Generally sin2 x+ cos2 x=1
∴ sin2 2θ=1−cos2 2θ

Surname 2
Replacing for sin2 2θ in the above equation, we obtain
cos2 2 θ+c os2 2 θ−1=2cos2 2 θ−1 as required
64.(1−sin¿¿ 2 2θ) ( 1+ tan2 2θ )=1 ¿
1−sin2 2 θ=cos2 2θ
Replacing for 1−sin2 2 θ in the equation
(cos ¿¿ 22 θ) ( 1+ tan2 2θ )=cos2 2 θ+cos2 2 θ tan2 2 θ ¿
But tan2 2 θ= sin2 2 θ
cos2 2 θ
cos2 2 θ+cos2 2 θ sin2 2 θ
cos2 2 θ =cos2 2θ+sin2 2 θ=1as required
Exercise 89-96: Use fundamental identities to find the values of the trigonometric functions for
the given conditions.
90. cot θ= 3
4 ∧cos θ<0
cot θ=cos θ
sinθ
cos θ∧sin θare both negative in the third quadrant to satisfy the condition cos θ< 0 and cot θ
is positive.
cot−1 3
4 =53.13°
In the third quadrant
53.13+180=233.13°
Exercise 51-74: Verify the identity by transforming the left-hand side into the right-hand side.
22. csc (−x) cos ( −x ) =−cot x
Since cosine is an even function and cosec is an odd function;
csc (−x )=−csc x and

Surname 3
cos ( −x ) =cos( x )
∴ cos ( x ) × −1
sin ( x) =−cos (x)
sin( x) =−cot ( x) as required.
Assignment 22
Exercise 39-46: Refer to the graph of y=sin x and y=cos x to find the exact values of x in the
interval [0, 4 π] that satisfy the equation.
44. cos x=−1
x=π , 3 π
Exercise 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the
equation.
10. y=cos(x− π
3 )
Considering the general form of a cosine function
a cos (bθ ±¿α )¿ where a is the amplitude∧α isthe phase angle,
Amplitude = 1
Phase shift = π
3
Period = 2 π
b = 2 π
1 =2 π

Surname 4
Sketch of the graph
Exercise 1-52: Find the period and sketch the graph of the equation. Show the asymptotes.
10. y=tan(x + π
2 )
Period = π
Phase shift: −π
2
Vertical asymptotes: x=−π +πn where nis an integer
Sketch of the graph

Surname 5
Exercise 1-8: Given the indicated parts of triangle ABC with γ=90°, find the exact values of the
remaining parts.
2. β=45° , b=35
a2+ b2=c2
Sine rule a
sin α = b
sin β
Hence, a
sin 45 = 35
sin 45 since α =β=45 ° because γ=90°
∴ a=35
c= √ a2+ b2= √ 352 +352=49.5
Exercise 7-18: Find the exact value.
8. (a) sin 210 °
210 °is in third quadrant, sine is negative in this quadrant hence sin 210 °=−sin 30 °=−0.5

Surname 6
(b)sin(−315 ° )
−315 ° is in the first quadrant where sine is positive hence sin(−315° )=sin 45° = √ 2
2 =0.7071
Work Cited
Bird, John. Higher Engineering Mathematics. Routlegde, 2010
Steward, James, et al. Precalculus: Mathematics for Calculus. Centgage Learning, 2013.