Advanced Trigonometry: Identities, Graphs, and Formulas

Math

11th Grade,
12th Grade

CCSS

NCTM

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| Attempts: 16 | Questions: 14 | Updated: Sep 13, 2025

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1) Rewrite the sum (or difference) as a product of sines and cosines of positive arguments: sin3x + sin9x?

Cos6x sin3x

Answer: 2 sin6x cos3xsin3x + sin 9xsin x + sin y = 2sin((x + y)/2) cos((x - y)/2) Sum to... Answer: 2 sin6x cos3xsin3x + sin 9xsin x + sin y = 2sin((x + y)/2) cos((x - y)/2) Sum to Product Formulae.sin3x + sin 9x = 2sin((3x + 9x)/2) cos((3x - 9x)/2)= 2 sin6x cos(-3x) Perform the operations and simplify.= 2 sin6x cos3x Even/Odd identities.

2 sin3x cos9x

Sin6x cos3x

When writing a sum or difference of sines as a product of sines and cosines, the formula to be used is 2sin((x + y)/2)cos((x - y)/2). In this case, sin3x + sin9x simplifies to 2 sin6x cos3x.

Explanation

When writing a sum or difference of sines as a product of sines and cosines, the formula to be used is 2sin((x + y)/2)cos((x - y)/2). In this case, sin3x + sin9x simplifies to 2 sin6x cos3x.

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About This Quiz

Explore the depths of trigonometric identities, formulas, graphing techniques, and inverse functions. This exercise enhances understanding of complex trigonometric concepts, crucial for advancing skills in mathematics and relevant fields.

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2) Rewrite the sum (or difference) as a product of sines and cosines of positive arguments. cos8x - cos10x

Sin8x cos10x

Cos8x + cos10x

Answer: 2 sin9x sinxcos8x - cos10xcos x - cos y = -2sin((x + y)/2) sin((x - y)/2) Sum to Product... Answer: 2 sin9x sinxcos8x - cos10xcos x - cos y = -2sin((x + y)/2) sin((x - y)/2) Sum to Product Formulae.cos8x - cos10x = -2sin((8x + 10x)/2) sin((8x - 10x)/2)= -2 sin9x sin(-x) Perform the operations and simplify.= -2 sin9x -sinx Even/Odd identities.= 2 sin9x sinx Simplify, a negative times a negative equals a positive.

Cos9x sinx

The correct answer is obtained by applying the Sum to Product Formulae for trigonometric functions, resulting in 2 sin9x sinx.

Explanation

The correct answer is obtained by applying the Sum to Product Formulae for trigonometric functions, resulting in 2 sin9x sinx.

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3) Write out the Product to Sum Formulas (for sin and cos ... there are four).

CosAcosB = sin(A + B) - sin(A - B)

SinAcosB = 1/2[sin(A + B) + sin(A - B)]cosAsinB = 1/2[sin(A + B) - sin(A - B)]cosAcosB = 1/2[cos(A +... SinAcosB = 1/2[sin(A + B) + sin(A - B)]cosAsinB = 1/2[sin(A + B) - sin(A - B)]cosAcosB = 1/2[cos(A + B) + cos(A - B)]sinAsinB = 1/2[cos(A - B) - cos(A + B)]

SinAcosB = cos(A + B) + cos(A - B)

CosAsinB = sin(A + B) + sin(A - B)

The Product to Sum Formulas are used to expand the product of two trigonometric functions into the sum of trigonometric functions. Each formula involves a combination of sine and cosine functions, either added or subtracted. The correct formulas represent the relationships between sine and cosine that hold true when multiplying two trigonometric functions.

Explanation

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4) Verify the identity: tan2x = (2tan x)/(1 - tan^2 x).

Tan2x = tan(x) * tan(x)

Tan2x = tan(x + x) Split the angle.= (tanx + tanx)/(1 - tan x tan x) Use the Sum and... Tan2x = tan(x + x) Split the angle.= (tanx + tanx)/(1 - tan x tan x) Use the Sum and Difference Formulae.= (2tanx)/(1 - tan^2 x) Simplify.

Tan2x = (tan x) ^ 2

Tan2x = 2tan x

The correct answer involves splitting the angle, using the Sum and Difference Formulae, and simplifying to achieve the desired identity.

Explanation

The correct answer involves splitting the angle, using the Sum and Difference Formulae, and simplifying to achieve the desired identity.

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5) Rewrite the expression cos^4 x in terms of the first power of cosine.

1/16(3 + 8cos2x + cos4x)

1/8(3 + 2cos2x + cos4x)

1/8(3 + 4cos2x + cos4x)cos^4 x(cos^2 x)^x = ((1 + cos^2 x)/2)^2 Split the power and use the Power Reduction... 1/8(3 + 4cos2x + cos4x)cos^4 x(cos^2 x)^x = ((1 + cos^2 x)/2)^2 Split the power and use the Power Reduction Formulae.= (1 + 2cos2x + cos^2 2x)/4 Square everything. = (1 + 2cos2x + ((1 + cos4x)/2))/4 Use the Power Reduction Formulae again.= (1/2(2) + 1/2(4cos2x) + 1/2(1 + cos4x))/4 Take out the 1/2. = 1/8(3 + 4cos2x + cos4x) Divide by 4.

1/4(3 + 4cos2x + cos4x)

The correct answer is obtained by applying the Power Reduction Formulae to split the power of cosine and simplify the expression step by step. Incorrect answers may result from errors in applying the formulas or misinterpretation of the question.

Explanation

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6) Rewrite the expression in terms of the first power of cosine: sin^2 x cos^2 x

1/10(1 - cos4x]

1/4(1 - cos4x)

1/6(1 - cos4x)

1/8(1 - cos4x)sin^2 x cos^2 x = ((1 - cos2x)/2)((1 + cos2x)/2) Power Reduction Formulae.= 1/2(1 - cos2x)1/2(1 + cos2x)... 1/8(1 - cos4x)sin^2 x cos^2 x = ((1 - cos2x)/2)((1 + cos2x)/2) Power Reduction Formulae.= 1/2(1 - cos2x)1/2(1 + cos2x) Take out the 1/2's.= 1/4(1 - cos2x)(1 + cos2x)= 1/4(1 - cos^2 2x) Multiply.= 1/4(1 - (1 - cos4x)/2) Power Reduction Formulae.= 1/4 - ((1 - cos4x)/8) Perform the operations and simplify.= 2/8 - ((1 - cos4x)/8)= (1 - cos4x)/8= 1/8(1 - cos4x)

The correct answer is derived using the power reduction formulae to simplify sin^2 x cos^2 x to 1/8(1 - cos4x). The incorrect answers deviate by using incorrect coefficients or skip steps in the simplification process.

Explanation

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7) Rewrite the product as a sum (or difference) of sines and cosines of positive arguments sin3x cos5x.

Sin8x + sin2x

1/2(sin8x - sin2x)sin3x cos5xsinAcosB = 1/2[sin(A + B) + sin(A - B)] Product to Sum Formulae. sin3x cos5x = 1/2[sin(3x... 1/2(sin8x - sin2x)sin3x cos5xsinAcosB = 1/2[sin(A + B) + sin(A - B)] Product to Sum Formulae. sin3x cos5x = 1/2[sin(3x + 5x) + sin(3x - 5x)]= 1/2(sin8x + sin(-2x)) Perform the operations and simplify.= 1/2 (sin8x - sin2x) Even/Odd Identities.

1/2(sin4x - sin6x)

1/2(sin7x - sin3x)

The correct way to rewrite sin3x cos5x as a sum of sines and cosines is to use the product to sum formulae, resulting in 1/2(sin8x - sin2x). This transforms the product into a combination of sine functions with positive arguments.

Explanation

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8) Rewrite the product as a sum (or difference) of sines and cosines of positive arguments: cos4x cos7x

1/2(sin3x - sin11x)

Answer: 1/2(cos3x + cos11x)cos4x cos7xcosAcosB = 1/2[cos(A - B) + cos(A + B) Product to Sum Formulae. cos4x cos7x =... Answer: 1/2(cos3x + cos11x)cos4x cos7xcosAcosB = 1/2[cos(A - B) + cos(A + B) Product to Sum Formulae. cos4x cos7x = 1/2[cos(4x - 7x) + cos(4x + 7x)]

1/2(sin3x + sin11x)

1/2(cos3x - cos11x)

To rewrite the product cos4x cos7x as a sum or difference of sines and cosines of positive arguments, we can use the product to sum formula. Therefore, cos4x cos7x = 1/2[cos(4x - 7x) + cos(4x + 7x)] = 1/2[cos(-3x) + cos(11x)] = 1/2[cos(3x) + cos(11x)] = 1/2(cos3x + cos11x).

Explanation

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9) Write out the Double Angle Formulas(there are 5).

Sin(2θ) = 2sin(θ)cos(θ) cos(2θ) = cos^2(θ) - sin^2(θ)tan(2θ) = (2tan(θ))/(1 - tan^2(θ)) csc(2θ) = 2csc(θ)cos(θ)sec(2θ) = sec^2(θ) - tan^2(θ).

Sin(2θ) = sin^2(θ) - cos^2(θ)cos(2θ) = 2cos(θ)sin(θ)tan(2θ) = tan^2(θ) - 1csc(2θ) = csc^2(θ) - cos^2(θ)

The Double Angle Formulas in trigonometry are used to express trigonometric functions of double angles in terms of functions of a single angle. The correct formulas are sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos^2(θ) - sin^2(θ), tan(2θ) = (2tan(θ))/(1 - tan^2(θ)), csc(2θ) = 2csc(θ)cos(θ), and sec(2θ) = sec^2(θ) - tan^2(θ). The incorrect answers provided do not accurately represent the Double Angle Formulas.

Explanation

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10) Graph one complete cycle of 4(cos 2x - ? /2).

4(cos 2x)

4(sin 2x)

4(cos2x−1/2)

4(cos x - 1/2)

The correct answer involves subtracting 1/2 within the cosine function, producing a specific graph pattern. The incorrect answers change either the constant value or the trigonometric function used, resulting in different graph shapes.

Explanation

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11) Write out the Sum and Difference Formulas (there are 6).

Sin(A+B)=sinAcosB+cosAsinB

Sin(A−B)=sinAcosB−cosAsinB

Cos(A+B)=cosAcosB−sinAsinB

Cos(A−B)=cosAcosB+sinAsinB

All of the above

The correct equation provided is y = 4cos⁡(2x), not the incorrect alternatives.

Explanation

The correct equation provided is y = 4cos⁡(2x), not the incorrect alternatives.

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12) Use the double angle formulae for the given values: cos2A, A = pi/4.

Cos⁡(2A)=cos⁡2A−sin⁡2A\cos(2A) = \cos^2 A - \sin^2 Acos(2A)=cos2A−sin2A → value = 000

Cos⁡(2A)=2cos⁡2A−1\cos(2A) = 2\cos^2 A - 1cos(2A)=2cos2A−1 → value = 000

Cos⁡(2A)=1−2sin⁡2A\cos(2A) = 1 - 2\sin^2 Acos(2A)=1−2sin2A → value = 000

Cos(2A)=cosA+sinA → value = 2\sqrt{2}2

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13) Find the equation to match:

Y = 5cos(3x)

Y = 2cos(4x)

Y = 4sin⁡(2x)

Y = 4cos ?x

The correct equation provided is y = 4cos⁡(2x), not the incorrect alternatives.

Explanation

The correct equation provided is y = 4cos⁡(2x), not the incorrect alternatives.

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14) Write the Half - Angle Formulas (For sin, cos, tan).

Cos(theta/2) = sin(theta)/cos(theta)

Tan(theta/2) = sin(theta - cos(theta))

Sin2A=±21−cosA, cos2A=±21+cosA

Sin(theta/2) = 2sin(theta)cos(theta)

The correct half-angle formulas for sin, cos, and tan involve square roots and are derived from the double angle formulas. Incorrect answers (a), (b), and (c) do not accurately represent the half-angle formulas for trigonometric functions.

Explanation

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