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Trigonometry Week 11 Test Grade Report

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Trigonometry

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Exam Practice

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Trigonometry Week 11 Test - Grade Report

Score:

90% (89.6 of 100 pts)

Submitted:

Jul 19 at 8:27pm

Question: 1

Grade: 1.0 / 4.0

Identify ALL equivalent forms of the expression.

36+36costsint−27sin2t36+36costsint−27sin2t

Select all that apply.

(6cost+3sint)26cost+3sint2

, (6cost+3sint)(6cost+3sint)6cost+3sint6cost+3sint (100%)

Solution

36+36costsint−27sin2t36+36costsint−27sin2t

= 36(cos2t+sin2t)+36costsint−27sin2t36cos2t+sin2t+36

costsint−27sin2t

Pythagorean

identity, cos2t+sin2t=1cos2t

+sin2t=1.

= 36cos2t+36costsint+(36−27)sin2t36cos2t+36costsint+

36−27sin2t

Expand and collect the like

terms.

= 36cos2t+36costsint+9sin2t36cos2t+36costsint+9sin2t

Simplify.

= (6cost+3sint)(6cost+3sint)6cost+3sint6cost+3sint

Factor.

= (6cost+3sint)26cost+3sint2

So, the correct expressions

are (6cost+3sint)(6cost+3sint)6cost+3sint6cost+3sint and (6cost+3sint)26cost+3sint2.

Question: 2

Grade: 0.0 / 4.0

True or false? 1+2cot2x+cot4xcsc2x1+2cot2x+cot4xcsc2x= cot2xcot2xis an identity.

True (0%)

Comment:

Solution

1+2cot2x+cot4xcsc2x1+2cot2x+cot4xcsc2x

(1+cot2x)(1+cot2x)csc2x1+cot2x1+cot2xcsc2x

Factor.

csc2x⋅csc2xcsc2xcsc2x⋅csc2xcsc2x

Substitute csc2xcsc2xfor 1+cot2x1+cot2x(Pythagorean

Identity).

csc2xcsc2x

Simplify.

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Since csc2xcsc2x≠ cot2xcot2xfor all values of x, the equation is not an identity.

Question: 3

Grade: 1.0 / 4.0

Find all solutions of cot24x−2cot4x+1=0cot24x−2cot4x+1=0 on [0,π).0,π.

Select all that apply.

x=x=5π165π16, x=x=9π169π16, x=x=π16π16, x=x=13π1613π16 (100%)

Solution

Since 0≤x<π0≤x<π, multiply by 44 to find the interval for 4x4x, 0≤4x<4π0≤4x<4π.

Let t=4xt=4x and solve the equation

cot2t−2cott+1=0cot2t−2cott+1=0 on [0,4π)0,4π.

Factoring yields

(cott−1)2=0cott−12=0

Ifcott−1=0cott−1=0, then cott=1cott=1.

Since cot x = 1/(tan x), solvetan(x)=1tan⁡x=1.

Consider that the reference angle

for 11is π4π4 for tangent and tangent is positive

in quadrants I and III.

Therefore, t=t=π4π4, t=t=5π45π4, t=t=9π49π4 , t=t=13π413π4, ...

Because t=4x,t=4x,4x=4x=π4π4, 5π45π4, 9π49π4, 13π413π4, ...

Therefore, x=x=π16π16, 5π165π16, 9π169π16, 13π1613π16 within 0≤x<π0≤x<π.

Question: 4

Grade: 1.0 / 4.0

Find all solutions of 6sin2x−2sinx−1=06sin2x−2sinx−1=0 on [0,2π).0,2π. Round to the

nearest thousandth.

Select all that apply.

x≈6.005x≈6.005, x≈3.420x≈3.420, x≈0.653x≈0.653, x≈2.489x≈2.489 (100%)

Solution

Let sinx=zsinx=z. Substitute.

6sin2x−2sinx−1=06sin2x−2sinx−1=0

6 z2−2z−1=06 z2−2z−1=0

This expression cannot be factored, so use the Quadratic Formula.

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Trigonometry Week 11 Test - Grade Report Score: 90% (89.6 of 100 pts) Submitted: Jul 19 at 8:27pm Question: 1 Grade: 1.0 / 4.0 Identify ALL equivalent forms of the expression. 36+36costsint−27sin2t36+36costsint−27sin2t Select all that apply. (6cost+3sint)26cost+3sint2 , (6cost+3sint)(6cost+3sint)6cost+3sint6cost+3sint (100%) Solution 36+36costsint−27sin2t36+36costsint−27sin2t Pythagorean = 36(cos2t+sin2t)+36costsint−27sin2t36cos2t+sin2t+36 identity, cos2t+sin2t=1cos2t costsint−27sin2t +sin2t=1. = 36cos2t+36costsint+(36−27)sin2t36cos2t+36costsint+ Expand and collect the like terms. 36−27sin2t = 36cos2t+36costsint+9sin2t36cos2t+36costsint+9sin2t Simplify. = (6cost+3sint)(6cost+3sint)6cost+3sint6cost+3sint Factor. = (6cost+3sint)26cost+3sint2 So, the correct expressions are (6cost+3sint)(6cost+3sint)6cost+3sint6cost+3sint and (6cost+3sint)26cost+3sint2. Question: 2 Grade: 0.0 / 4.0 True or false? 1+2cot2x+cot4xcsc2x1+2cot2x+cot4xcsc2x= cot2xcot2xis an identity. True (0%) Comment: Solution 1+2cot2x+cot4xcsc2x1+2cot2x+cot4xcsc2x == (1+cot2x)(1+cot2x)csc2x1+cot2x1+cot2xcsc2x Facto == csc2x⋅csc2xcsc2xcsc2x⋅csc2xcsc2x Subst Identi == csc2xcsc2x Simp Since csc2xcsc2x≠ cot2xcot2xfor all values of x, the equation is not an identity. Question: 3 Grade: 1.0 / 4.0 Find all solutions of cot24x−2cot4x+1=0cot24x−2cot4x+1=0 on [0,π).0,π. Select all that apply. x=x=5π165π16, x=x=9π169π16, x=x=π16π16, x=x=13π1613π16 (100%) Solution Since 0? ...
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Tags: trigonometry positive tangent Coterminal Angles Decimal approximation positive in quadrants