Precalculus : Trigonometric Functions

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #131 : Trigonometric Functions

Simplify.

\(\displaystyle 1-\tan x(\csc x)\)

Possible Answers:

\(\displaystyle 1-\sec x\)

\(\displaystyle 2\)

\(\displaystyle \tan^2 x\)

None of these answers are correct.

\(\displaystyle \sin^2x\)

Correct answer:

\(\displaystyle 1-\sec x\)

Explanation:

\(\displaystyle \frac{\sin x}{\cos x}=\tan x\)

\(\displaystyle \frac{1}{\sin x}=\csc x\)

 

Given these identities...

\(\displaystyle 1-\tan x(\csc x)\)

\(\displaystyle 1- \frac{\sin x}{\cos x}(\frac{1}{\sin x})\)

\(\displaystyle 1-\frac{1}{\cos x}\)

\(\displaystyle 1- \sec x\)

Example Question #132 : Trigonometric Functions

Simplify \(\displaystyle \frac{\sin (x)}{1-\cos (x)}-\cot(x)\) completely. 

Possible Answers:

\(\displaystyle -\sec (x)\)

\(\displaystyle -\tan (x)\)

\(\displaystyle 1-\tan(x)\)

\(\displaystyle \csc (x)\)

\(\displaystyle \cos (x)-\sin(x)\)

Correct answer:

\(\displaystyle \csc (x)\)

Explanation:

First simplify the fraction 

\(\displaystyle \frac{\sin (x)}{1-\cos (x)}\) 

by multiplying it by its conjugate

\(\displaystyle \frac{1+\cos (x)}{1+\cos (x)}\).

After doing so, continue simplying:

\(\displaystyle \\ \frac{\sin(x)(1+\cos(x))}{(1-\cos(x))(1+\cos (x))}- \cot(x)\\ \\=\frac{\sin(x)(1+\cos(x))}{1-\cos^2(x)}-\cot(x)\\ \\=\frac{\sin(x)(1+\cos(x))}{\sin^2(x)}-\cot(x)\\ \\=\frac{1+\cos(x)}{\sin(x)}-\frac{\cos(x)}{\sin(x)}\\ \\=\frac{1}{\sin(x)}\\ \\=\csc (x)\) 

Example Question #133 : Trigonometric Functions

Fully simplify.

Simplify:

\(\displaystyle \frac{(\sin ^2 x+ \cos^2 x)}{(1+ \cot^2 x)}\)

Possible Answers:

\(\displaystyle \cos^2 x\)

\(\displaystyle \sin ^2 x\)

\(\displaystyle 1\)

\(\displaystyle \sin x + \cos x\)

None of these answers are correct.

Correct answer:

\(\displaystyle \sin ^2 x\)

Explanation:

\(\displaystyle \sin^2 x + \cos^2 x = 1\)

\(\displaystyle 1+ \cot^2x= \csc^2 x\)

 

Given the above identities:

 

\(\displaystyle \frac{(\sin ^2 x+ \cos^2 x)}{(1+ \cot^2 x)}=\frac{1}{\csc^2 x}=\sin^2x\)

 

Example Question #134 : Trigonometric Functions

Simplify:

\(\displaystyle \frac{1-\cos ^2x}{\sin ^2x}+(\sin ^2x + \cos^2 x)\)

Possible Answers:

\(\displaystyle 2\)

\(\displaystyle 1\)

None of these answers are correct.

\(\displaystyle 1+2\sin^2x\)

\(\displaystyle 1-2\cos^2x\)

Correct answer:

\(\displaystyle 2\)

Explanation:

\(\displaystyle \sin^2x+\cos^2x=1\)

and

\(\displaystyle \sin^2x=1-\cos^2x\)

 

Therefore...

\(\displaystyle \frac{1-\cos ^2x}{\sin ^2x}+(\sin ^2x + \cos^2 x)\)

\(\displaystyle =\frac{\sin^2x}{\sin^2x}+(\sin^2x+\cos^2x)\)

\(\displaystyle =1+(\sin ^2x+\cos ^2x)\)

\(\displaystyle =1+1=2\)

 

Example Question #131 : Trigonometric Functions

Find the exact value

\(\displaystyle sin\left[2tan^{-1}\left(\frac{3}{4}\right)\right]\).

Possible Answers:

\(\displaystyle \frac{6}{5}\)

\(\displaystyle \frac{12}{25}\)

\(\displaystyle \frac{24}{25}\)

\(\displaystyle \frac{9}{8}\)

Correct answer:

\(\displaystyle \frac{24}{25}\)

Explanation:

By the double angle formula

\(\displaystyle {\right }sin(2x)=2sin(x)cos(x)\)

\(\displaystyle sin\left[2tan^{-1}\left(\frac{3}{4}\right)\right]=2sin\left[tan^{-1}\left(\frac{3}{4}\right)\right]cos\left[tan^{-1}\left(\frac{3}{4}\right)\right]\)

\(\displaystyle =2\left(\frac{3}{5}\right)\left(\frac{4}{5}\right)=\frac{24}{25}\)

Example Question #22 : Fundamental Trigonometric Identities

Find the exact value

\(\displaystyle sin\left[2tan^{-1}\left(\frac{8}{15}\right)\right]\).

Possible Answers:

\(\displaystyle \frac{16}{289}\)

\(\displaystyle \frac{16}{17}\)

\(\displaystyle \frac{240}{289}\)

\(\displaystyle \frac{120}{289}\)

Correct answer:

\(\displaystyle \frac{240}{289}\)

Explanation:

By the double angle formula

\(\displaystyle {}sin(2x)=2sin(x)cos(x)\)

\(\displaystyle sin\left[2tan^{-1}\left(\frac{8}{15}\right)\right]=2sin\left[tan^{-1}\left(\frac{8}{15}\right)\right]cos\left[tan^{-1}\left(\frac{8}{15}\right)\right]\)

\(\displaystyle =2\left(\frac{8}{17}\right)\left(\frac{15}{17}\right)=\frac{240}{289}\)

Example Question #23 : Fundamental Trigonometric Identities

Find the exact value

\(\displaystyle cos\left[2tan^{-1}\left(\frac{5}{12}\right)\right]\).

Possible Answers:

\(\displaystyle \frac{50}{169}\)

\(\displaystyle \frac{7}{13}\)

\(\displaystyle \frac{119}{169}\)

\(\displaystyle -\frac{119}{169}\)

Correct answer:

\(\displaystyle \frac{119}{169}\)

Explanation:

By the double-angula formula for cosine

\(\displaystyle cos(2x)=cos^2(x)-sin^2(x)\)

For this problem

\(\displaystyle cos\left[2tan^{-1}\left(\frac{5}{12}\right)\right]=cos^2\left[tan^{-1}\left(\frac{5}{12}\right)\right]-sin^2\left[tan^{-1}\left(\frac{5}{12}\right)\right]\)

\(\displaystyle =\left(\frac{12}{13}\right)^2-\left(\frac{5}{13}\right)^2=\frac{119}{169}\)

Example Question #22 : Fundamental Trigonometric Identities

Find the exact value

\(\displaystyle sin\left[2sin^{-1}\left(\frac{12}{37}\right)\right]\).

Possible Answers:

\(\displaystyle \frac{840}{37}\)

\(\displaystyle \frac{420}{1369}\)

\(\displaystyle \frac{840}{1369}\)

\(\displaystyle \frac{24}{37}\)

Correct answer:

\(\displaystyle \frac{840}{1369}\)

Explanation:

By the double-angle formula for the sine function

\(\displaystyle sin(2x)=2sin(x)cos(x)\)

we have

\(\displaystyle x=sin^{-1}\left( \frac{12}{37}\right)\)

thus the double angle formula becomes,

\(\displaystyle sin\left[2sin^{-1}\left(\frac{12}{37}\right)\right]=2sin\left[sin^{-1}\left(12/37\right)\right]cos\left[sin^{-1}\left(12/37\right)\right]\)

\(\displaystyle =2\left(\frac{12}{37}\right)\left(\frac{35}{37}\right)=\frac{840}{1369}\)

Example Question #132 : Trigonometric Functions

If \(\displaystyle \theta=X\), which of the following best represents \(\displaystyle sin(2\theta)\)?

Possible Answers:

\(\displaystyle sinXcosX\)

\(\displaystyle cos^2X-sin^2X\)

\(\displaystyle 2sinXcosX\)

\(\displaystyle sin^2X\)

\(\displaystyle 1-cos^2X\)

Correct answer:

\(\displaystyle 2sinXcosX\)

Explanation:

The expression \(\displaystyle sin(2\theta)\) is a double angle identity that can also be rewritten as:

\(\displaystyle 2sin(\theta) cos(\theta)\)

Replace the value of theta for \(\displaystyle \theta=X\).

The correct answer is: \(\displaystyle 2sin(X) cos(X)\)

Example Question #1504 : Pre Calculus

Which expression is equivalent to \(\displaystyle \cos(6x)\) ?

Possible Answers:

\(\displaystyle 1 - 6 \sin ^ 2 x\)

\(\displaystyle 6 \cos ^ 2 x - 1\)

\(\displaystyle \cos ^ 2 x - \sin ^ 2 x\)

\(\displaystyle 1- 2 \sin^2 (3x)\)

\(\displaystyle 3 \cos ^ 2 (2x ) - 1\)

Correct answer:

\(\displaystyle 1- 2 \sin^2 (3x)\)

Explanation:

The relevant trigonometric identity is:

\(\displaystyle \cos (2u) = \cos ^ 2 u - \sin ^ 2 u = 2 \cos ^ 2 u - 1 = 1 - 2 \sin ^ 2 u\)

In this case, "u" is \(\displaystyle 3x\) since \(\displaystyle 2*3x = 6x\).

The only one that actually follows this is \(\displaystyle 1- 2 \sin^2 (3x)\)

 

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