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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 #111 : Trigonometry

If $sin\ x\times cos\ x=\frac{1}{2}$ , what is the value of $(sin\ x+cos\ x)^2$ ?

Possible Answers:

1.5

2.5

Correct answer:

Explanation:

$(sin\ x+cos\ x)^2=sin^2x+cos^2x+2sin\x cos\ x$

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

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Example Question #33 : Trigonometric Functions And Graphs

If $sin^2x-cos^2x=\frac{1}{2}$ , give the value of cos^4x-sin^4x .

Possible Answers:

$\frac{1}{2}$

$-\frac{1}{2}$

Correct answer:

$-\frac{1}{2}$

Explanation:

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

We know that cos^2x+sin^2x=1 .

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

$=(cos^2x-sin^2x)\times 1=-(sin^2x-cos^2x)=-\frac{1}{2}$

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Example Question #12 : Simplifying Trigonometric Functions

Simplify:

A=(cos^4x-sin^4x)(1+tan^2x)+tan^2x

Possible Answers:

1.5

2.5

0.5

Correct answer:

Explanation:

A=(cos^4x-sin^4x)(1+tan^2x)+tan^2x

= (cos^2x-sin^2x)(cos^2x+sin^2x)(1+tan^2x)+tan^2x

We know that cos^2x+sin^2x=1 .

Then we can write:

A= (cos^2x-sin^2x)(1+tan^2x)+tan^2x

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

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

$\Rightarrow A=\frac{(cos^2x-sin^2x)}{cos^2x}+tan^2x$

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

=1-tan^2x+tan^2x

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Example Question #13 : Simplifying Trigonometric Functions

Which of the following is equal to $\sin x\tan x$ ?

Possible Answers:

$\frac{\sin^2x}{\cos x}$

$\sec x \csc x$

$\frac{\cos^2x}{\sin x}$

$\cos x \cot x$

$\sin x \csc x$

Correct answer:

$\frac{\sin^2x}{\cos x}$

Explanation:

Break $\tan x$ apart: $\tan x =\frac{\sin x }{\cos x}$ .

This means that $\sin x\tan x=\sin x *\frac{\sin x}{\cos x}$ or $\frac{\sin^2 x}{\cos x}$

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Example Question #12 : Simplifying Trigonometric Functions

Simplify: $\frac{sin(\theta)}{tan(\theta)}-\frac{csc(\theta)}{sec(\theta)}$

Possible Answers:

$cos(\theta)+cot(\theta)$

$sec(\theta)-tan(\theta)$

$cos(\theta)-sin(\theta)cos(\theta)$

$cos(\theta)cot(\theta)-cot(\theta)$

$cos(\theta)-cot(\theta)$

Correct answer:

$cos(\theta)-cot(\theta)$

Explanation:

Rewrite $\frac{sin(\theta)}{tan(\theta)}-\frac{csc(\theta)}{sec(\theta)}$ in terms of sines and cosines.

$\frac{sin(\theta)}{tan(\theta)}-\frac{csc(\theta)}{sec(\theta)} = \frac{sin(\theta)}{\frac{sin(\theta)}{cos(\theta)}}-\frac{\frac{1}{sin(\theta)}}{\frac{1}{cos(\theta)}}$

Simplify the complex fractions.

$\frac{sin(\theta)}{\frac{sin(\theta)}{cos(\theta)}}-\frac{\frac{1}{sin(\theta)}}{\frac{1}{cos(\theta)}}=sin(\theta)\cdot \frac{cos(\theta)}{sin(\theta)}-\frac{1}{sin(\theta)}\cdot \frac{cos(\theta)}{1}$

$=cos(\theta)-\frac{cos(\theta)}{sin(\theta)}= cos(\theta)-cot(\theta)$

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Example Question #14 : Simplifying Trigonometric Functions

Simplify the following expression:

$sec^2xcot^2x \ - sec^2x + tan^2x$

Possible Answers:

tan^2x

cot^2x

$cos \ 2x$

cos^2x

Correct answer:

cot^2x

Explanation:

We will first invoke the appropriate ratio for cotangent, and then use pythagorean identities to simplify the expression:

$sec^2xcot^2x \ - sec^2x + tan^2x$

$= \frac{1}{cos^2x} * \frac{cos^2}{sin^2x} - (sec^2 x - tan^2x){}$

$= \frac{1}{sin^2x} - (sec^2x - tan^2x){}$

$= csc^2x - 1{}$

since

$sec^2x - tan^2x = 1; \ csc\ x = \frac{1}{sin \ x}{}$

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Example Question #15 : Simplifying Trigonometric Functions

Simplify the trigonometric expression.

$\frac{\tan(\theta)\csc(\theta)}{\sec(\theta)}$

Possible Answers:

$\cos(\theta)$

$\sec\theta$

$\tan(\theta)$

Correct answer:

Explanation:

Using basic trigonometric identities, we can simplify the problem to

$\frac{\frac{\sin(\theta)}{\cos(\theta)}*\frac{1}{\sin(\theta)}}{\frac{1}{\cos(\theta)}}$ .

We can cancel the sine in the numerator and the one over cosine cancels on top and bottom, leaving us with 1.

$\frac{\frac{\sin(\theta)}{\cos(\theta)}*\frac{1}{\sin(\theta)}}{\frac{1}{\cos(\theta)}}=\frac{\frac{1}{\cos(\theta)}}{\frac{1}{\cos(\theta)}}=\frac{1}{\cos(\theta)}\cdot \frac{\cos(\theta)}{1}=1$

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Example Question #16 : Simplifying Trigonometric Functions

Simplify.

$\frac{\sin x\tan x}{\cos x}$

Possible Answers:

$2\sin x$

$\sin x\cos x$

$\tan^2 x$

$\tan x$

Correct answer:

$\tan^2 x$

Explanation:

First, put everything in terms of sine and cosine:

$\frac{\sin x\tan x}{\cos x} \rightarrow \frac{\sin x\tfrac{\sin x}{\cos x}}{\cos x}\rightarrow \frac{\sin^2 x}{\cos^2 x}$

This simplifies to $\tan^2 x$ .

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Example Question #17 : Simplifying Trigonometric Functions

$\small \sec x*\sin x* \cot x$ If you simplify this equation. What might the new expression be?

Possible Answers:

$\small \sec x$

$\small \cos x$

$\small \tan x$

$\small \sin x$

Correct answer:

Explanation:

You should know two things. $\small \sec x = \frac{1}{\cos x}$ and $\small \cot x = \frac{\cos x}{\sin x}$ . If you place these to the equation, you will get $\small \frac{1}{\cos x}*\sin x*\frac{\cos x}{\sin x}$ . After you get rid of the same values, which are in the denominator and numerator, you will get 1.

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Example Question #21 : Simplifying Trigonometric Functions

Which of the following is equivalent to

$\tan\theta\csc\theta$ ?

Possible Answers:

$\sin\theta$

$\tan\theta$

$\csc\theta$

$\sec\theta$

$\cos\theta$

Correct answer:

$\sec\theta$

Explanation:

In order to evaluate this expression, rewrite the trigonometric identity in terms of sines and cosines. The tangent is equal to the sine over the cosine and the cosecant is the reciprocal of the sine; thus, we can write the following:

$\tan(\theta)\csc(\theta)= \frac{\sin(\theta)}{\cos(\theta)}\times\frac{1}{\sin(\theta)}$

Now, can simplify. Notice that the sine terms cancel each other out.

$\frac{\sin(\theta)}{\cos(\theta)}\times\frac{1}{\sin(\theta)}= \frac{1}{\cos(\theta)}$

Remember, that the reciprocal of the cosine is the secant.

$\frac{1}{\cos(\theta)}= \sec(\theta)$

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Trigonometry : Trigonometry

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #111 : Trigonometry

Example Question #33 : Trigonometric Functions And Graphs

Example Question #12 : Simplifying Trigonometric Functions

Example Question #13 : Simplifying Trigonometric Functions

Example Question #12 : Simplifying Trigonometric Functions

Example Question #14 : Simplifying Trigonometric Functions

Example Question #15 : Simplifying Trigonometric Functions

Example Question #16 : Simplifying Trigonometric Functions

Example Question #17 : Simplifying Trigonometric Functions

Example Question #21 : Simplifying Trigonometric Functions

All Trigonometry Resources

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