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Trigonometry : Understanding Trigonometric Equations

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #6 : Factoring Trigonometric Equations

Factor the following expression

$sin^{2n}(x)+2sin^n(x)+3$ where is assumed to be a positive integer.

Possible Answers:

(sin^nx+1)(sin^nx+2)

(sin^nx+1)(sin^nx-2)

(2sin^nx+1)(sin^nx-2)

We cannot factor the above expression.

(sin^nx-1)(sin^nx-2)

Correct answer:

We cannot factor the above expression.

Explanation:

Letting sin^n(x)=X , we have the equivalent expression:

X^2+2X+3 .

We cant factor X^2+2X+3 since $X^2+2X+3=(x+1)^2+2 >0 \forall \quadx\in\mathbb{R}$ .

This shows that we cannot factor the above expression.

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Example Question #3 : Factoring Trigonometric Equations

Factor

$1-cos^{11}x$

Possible Answers:

$(1-cosx)(cos^{10}x+cos^{9}x+...cosx-1)$

$(1-cosx)(cos^{10}x+cos^{9}x+...cosx+1)$

$(1-cosx)(cos^{11}x+cos^{9}x+...cosx+1)$

$(-1-cosx)(cos^{10}x+cos^{9}x+...cosx+1)$

$(2-cosx)(cos^{10}x+cos^{9}x+...cosx+1)$

Correct answer:

$(1-cosx)(cos^{10}x+cos^{9}x+...cosx+1)$

Explanation:

We first note that we have:

$1-a^{11}=(1-a)(a^{10}+a^9+...a+1)$

Then taking a=cos(x) , we have the result.

$(1-cosx)(cos^{10}x+cos^{9}x+...cosx+1)$

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Example Question #4 : Factoring Trigonometric Equations

Find a simple expression for the following :

$cos^{2n}x-cos^{2n+2} x$

Possible Answers:

$-cos^{2n}xsin^{2}x$

3sinx

$cos^{2n}xsin^{2}x$

$cos^{2n}x (1-sin^{2}x)$

$2cos^{2n}xsin^{2}x$

Correct answer:

$cos^{2n}xsin^{2}x$

Explanation:

First of all we know that :

cos^2x+sin^2x=1 and this gives:

$1-cos^2{x}=sin^{2}x$ .

Now we need to see that: $cos^{2n}x-cos^{2n+2} x$ can be written as

$cos^{2n}x(1-cos^2x)$ and since $1-cos^2{x}=sin^{2}x$

we have then:

$cos^{2n}x(1-cos^2x)=cos^{2n}xsin^2x$ .

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Example Question #21 : Trigonometric Equations

What is a simple expression for the formula:

$cos^{3}x-sin^{3}x$

Possible Answers:

(cosx-sinx)(1+cosx sinx)

(cosx-sinx)(2-cosx sinx)

2(cosx-sinx)(1+cosx sinx)

(cosx+sinx)(1+cosx sinx)

(cosx-sinx)(1+3cosx sinx)

Correct answer:

(cosx-sinx)(1+cosx sinx)

Explanation:

From the expression :

a^3-b^3=(a-b)(a^2+ab+b^2)

we have:

$cos^{3}x-sin^{3}x=(cosx-sinx)(cos^2x+cosxsinx+sin^2x)$

Now since we know that :

cos^2x+sin^2x=1 . This expression becomes:

$cos^{3}x-sin^{3}x=(cosx-sinx)(1+cosxsinx)$ .

This is what we need to show.

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Example Question #22 : Trigonometric Equations

Factor: (sin^4x-cos^4x)

Possible Answers:

(sin^2x+cos^2x)(sinx+cosx)

(sinx-cosx)(sinx+cosx)

-(sin^2x+cos^2x)(sinx-cosx)

(sin^2x-cos^2x)(sin^2x+cos^2x)

Correct answer:

(sinx-cosx)(sinx+cosx)

Explanation:

Step 1: Recall the difference of squares (or powers of four) formula:

x^4-y^4=(x^2+y^2)(x^2-y^2)=(x^2+y^2)(x-y)(x+y)

Step 2: Factor the question:

(sin^4x-cos^4x)=(sin^2x+cos^2x)(sin^2x-cos^2x)

Factor more:

(sin^2x+cos^2x)(sinx+cosx)(sinx-cosx)

Step 3: Recall a trigonometric identity:

sin^2x+cos^2x=1 .. Replace this

Final Answer:

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Trigonometry : Understanding Trigonometric Equations

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #6 : Factoring Trigonometric Equations

Example Question #3 : Factoring Trigonometric Equations

Example Question #4 : Factoring Trigonometric Equations

Example Question #21 : Trigonometric Equations

Example Question #22 : Trigonometric Equations

All Trigonometry Resources

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