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Trigonometry : Trigonometry

Study concepts, example questions & explanations for Trigonometry

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6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : Simplifying Trigonometric Functions

Simplify the following trigonometric function in fraction form:

$cos^{2}(45^{\circ})-sin(30^{\circ})$

Possible Answers:

1/2

1/3

Correct answer:

Explanation:

To determine the value of the expression, you must know the following trigonometric values:

$cos(45^{\circ}) = \frac{\sqrt{2}}{2}$

$sin (30^{\circ})=\frac{1}{2}$

Replacing these values, we get:

$cos^{2}(45^{\circ})-sin(30^{\circ})$

$(\frac{\sqrt{2}}{2})^{2}-\frac{1}{2}$

$\frac{2}{4}-\frac{1}{2}=\frac{1}{2}-\frac{1}{2}=0$

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Example Question #3 : Simplifying Trigonometric Functions

$tan (x)*sin(x) + sec(x)*(cos x)^{2} =$

Possible Answers:

sec (x)

csc (x)

cos (x)* sin (x)

$(cos x)^{2}$

tan (x)

Correct answer:

sec (x)

Explanation:

$tan (x)*sin(x) + sec(x)*(cos x)^{2} =$

$\frac{sin (x)}{cos (x)}* sin(x) + \frac{1}{cos (x)}*(cos x)^{2} =$

$\frac{(sin x)^{2} + (cos x)^{2}}{cos (x)} = \frac{1}{cos (x)} = sec (x)$

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Example Question #4 : Simplifying Trigonometric Functions

Simplify the following expression:

$A=cos(\frac{3\pi}{2}+x)+sin (\pi+x)+cos(\frac{\pi}{2}+x)+sin(\pi-x)$

Possible Answers:

A=-1

$A=-sin\ x$

A=1

A=0

$A=sin\ x$

Correct answer:

A=0

Explanation:

We need to use the following identities:

$cos(\frac{3\pi}{2}+x)=sin\ x$

$sin (\pi+x)=-sin\ x$

$cos(\frac{\pi}{2}+x)=-sin\ x$

$sin(\pi-x)=sin\ x$

Use these to simplify the expression as follows:

$A=cos(\frac{3\pi}{2}+x)+sin (\pi+x)+cos(\frac{\pi}{2}+x)+sin(\pi-x)$

$\Rightarrow A=sin\ x-sin\ x-sin\ x+sin\ x = 0$

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Example Question #5 : Simplifying Trigonometric Functions

Give the value of :

$A=\frac{2sin(\frac{\pi}{2})+4cos(\frac{\pi}{2})}{4sin(\frac{\pi}{2})-cos(\frac{\pi}{2})}$

Possible Answers:

A=0

A=1.5

A=1

A=2

A=0.5

Correct answer:

A=0.5

Explanation:

$sin(\frac{\pi}{2})=1$

$cos(\frac{\pi}{2})=0$

Plug these values in:

$A=\frac{2sin(\frac{\pi}{2})+4cos(\frac{\pi}{2})}{4sin(\frac{\pi}{2})-cos(\frac{\pi}{2})}=\frac{2\times 1+4\times 0}{4\times 1-0}=\frac{2}{4}=\frac{1}{2}=0.5$

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Example Question #3 : Simplifying Trigonometric Functions

If $tan\ x = \frac{1}{4}$ , solve for

$D=\frac{sin\ x}{sin\ x-cos\ x}+\frac{sin\ x+cos\ x}{cos\ x}$

Possible Answers:

$D=\frac{11}{12}$

$D=\frac{7}{12}$

$D=-\frac{11}{12}$

$D=\frac{1}{12}$

$D=\frac{5}{12}$

Correct answer:

$D=\frac{11}{12}$

Explanation:

$tan\ x=\frac{1}{4}\Rightarrow \frac{sin\ x}{cos\ x}=\frac{1}{4}\Rightarrow cos\ x=4sin\ x$

Substitute $cos\ x=4sin \ x$ into the expression:

$D=\frac{sin\ x}{sin\ x-cos\ x}+\frac{sin\ x+cos\ x}{cos\ x}=\frac{sin\ x}{sin\ x-4sin\ x}+\frac{sin\ x+4sin\ x}{4sin\ x}$

$\Rightarrow D=\frac{sin\ x}{-3sin\ x}+\frac{5sin\ x}{4sin\ x}$

$\Rightarrow D=-\frac{1}{3}+\frac{5}{4}=\frac{-4+15}{12}=\frac{11}{12}$

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Example Question #2 : Simplifying Trigonometric Functions

If $cot\ x= 3$ , give the value of :

$D=\frac{cos^2x}{(sin\ x)(cos\ x)}-\frac{sin\ x+cos\ x}{sin\ x}$

Possible Answers:

D=-2

D=-1

D=0

D=1

D=2

Correct answer:

D=-1

Explanation:

$cot\ x=3\Rightarrow \frac{cos\ x}{sin\ x}=3\Rightarrow cos\ x= 3sin\ x$

Now substitute $cos\ x=3sin\ x$ into the expression:

$D=\frac{cos^2x}{(sin\ x)(cos\ x)}-\frac{sin\ x+cos\ x}{sin\ x}=\frac{(3sin\ x)^2}{(sin\ x)(3sin\ x)}-\frac{sin\ x+3sin\ x}{sin\ x}$

$\Rightarrow D=\frac{9sin^2x}{3sin^2x}-\frac{4sin\ x}{sin\ x}$

$\Rightarrow D=\frac{9}{3}-4=3-4=-1$

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Example Question #3 : Simplifying Trigonometric Functions

Simplify the following expression:

$A=\frac{sin^2x}{1+tan^2x}-\frac{cos^2x}{1+cot^2x}+1$

Possible Answers:

A=-2

A=1

A=2

A=-1

A=0

Correct answer:

A=1

Explanation:

We need to use the following identitities:

$\frac{1}{cos ^2x}=1+tan^2x$

$\frac{1}{sin^2x}=1+cot^2x$

Now substitute them into the expression:

$A=\frac{sin^2x}{1+tan^2x}-\frac{cos^2x}{1+cot^2x}+1=\frac{sin^2x}{\frac{1}{cos^2x}}-\frac{cos^2x}{\frac{1}{sin^2x}}+1$

$\Rightarrow A=(sin^2x)(cos^2x)-(sin^2x)(cos^2x)+1=0+1=1$

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Example Question #1 : Simplifying Trigonometric Functions

If $cos\ x=\frac{1}{3}$ and $sin\ x\neq 0$ , give the value of $\frac{sin\ x+sin\ 2x}{sin\ x-sin\ 2x}$ .

Possible Answers:

Correct answer:

Explanation:

Based on the double angle formula we have, $sin\ 2x=2(sin\ x)(cos\ x)$ .

$\frac{sin\ x+sin\ 2x}{sin\ x-sin\ 2x}=\frac{sin\ x+2(sin\ x)(cos\ x)}{sin\ x-2(sin\ x)(cos\ x)}$

$=\frac{sinx(1+2cos\ x )}{sinx(1-2cos\ x )}=\frac{1+2cos\ x}{1-2cos\ x}$

$=\frac{1+2(\frac{1}{3})}{1-2(\frac{1}{3})}=\frac{\frac{3+2}{3}}{\frac{3-2}{3}}=5$

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Example Question #101 : Trigonometry

If $x=\frac{\pi}{3}$ , give the value of $\frac{sin\ x}{sin\ x+cos\ 2x}$ .

Possible Answers:

$\frac{3+\sqrt{3}}{2}$

$\frac{3-\sqrt{2}}{2}$

$\frac{3+\sqrt{2}}{2}$

$\frac{3-\sqrt{3}}{2}$

Correct answer:

$\frac{3+\sqrt{3}}{2}$

Explanation:

$x=\frac{\pi}{3}\Rightarrow sin\ x=sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$

Now we can write:

$x=\frac{\pi}{3}\Rightarrow 2x=2(\frac{\pi}{3})=\frac{2\pi}{3}\Rightarrow cos\ 2x=cos(\frac{2\pi}{3})$

$=cos(\pi-\frac{\pi}{3})$

$=-cos (\frac{\pi}{3})$

$=-\frac{1}{2}$

Now we can substitute the values:

$\frac{sin\ x}{sin\ x+cos\ 2x}=\frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}-\frac{1}{2}}=\frac{\sqrt{3}}{\sqrt{3}-1}=\frac{\sqrt{3}}{\sqrt{3}-1}\times \frac{{\sqrt{3}+1}}{{\sqrt{3}+1}}=\frac{3+\sqrt{3}}{2}$

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Example Question #101 : Trigonometry

If $cot\ 34^{\circ}=1.5$ , give the value of $\frac{2sin\ 326^{\circ}+3sin\ 56^{\circ}}{cos\ 304^{\circ}}$ .

Possible Answers:

1.5

0.5

2.5

Correct answer:

2.5

Explanation:

$326^{\circ}=360^{\circ}-34^{\circ}\Rightarrow sin\ 326^{\circ}=sin(360^{\circ}-34^{\circ})\Rightarrow sin\ 326^{\circ}=-sin\ 34^{\circ}$

$56^{\circ}=90^{\circ}-34^{\circ}\Rightarrow sin\ 56^{\circ}=sin(90^{\circ}-34^{\circ})\Rightarrow sin\ 56^{\circ}=cos\ 34^{\circ}$

$304^{\circ}=270^{\circ}+34^{\circ}\Rightarrow cos\ 304^{\circ}=cos(270^{\circ}+34^{\circ})\Rightarrow cos\ 304^{\circ}=sin\ 34^{\circ}$

Now we can simplify the expression as follows:

$\frac{2sin\ 326^{\circ}+3sin\ 56^{\circ}}{cos\ 304^{\circ}}=\frac{-2sin\ 34^{\circ}+3cos\ 34^{\circ}}{sin\ 34^{\circ}}$

$=-2+3cot\ 34^{\circ}=-2+3\times 1.5=-2+4.5=2.5$

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Trigonometry : Trigonometry

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #2 : Simplifying Trigonometric Functions

Example Question #3 : Simplifying Trigonometric Functions

Example Question #4 : Simplifying Trigonometric Functions

Example Question #5 : Simplifying Trigonometric Functions

Example Question #3 : Simplifying Trigonometric Functions

Example Question #2 : Simplifying Trigonometric Functions

Example Question #3 : Simplifying Trigonometric Functions

Example Question #1 : Simplifying Trigonometric Functions

Example Question #101 : Trigonometry

Example Question #101 : Trigonometry

All Trigonometry Resources

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