Identities

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Worksheet Identities

1.

Use the figure below to find the exact values of the following.

\(\sin(2\alpha) =\)

\(\cos(2\alpha) =\)

\(\tan(2\alpha) =\)

Answer 1.

\(\frac{120}{169}\)

Answer 2.

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

Answer 3.

\(\frac{-120}{119}\)

2.

Simplify: \(\cos^2(24^\circ)-\sin^2(24^\circ)\)

\(\displaystyle \sin(48^\circ)\)
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\(\displaystyle \tan(48^\circ)\)
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\(\displaystyle \sin(12^\circ)\)
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\(\displaystyle \cos(48^\circ)\)
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\(\displaystyle \tan(12^\circ)\)
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\(\displaystyle \cos(12^\circ)\)
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None of the above
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Simplify: \(2\cos^2(78^\circ)-1\)

\(\displaystyle \tan(39^\circ)\)
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\(\displaystyle \sin(156^\circ)\)
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\(\displaystyle \sin(39^\circ)\)
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\(\displaystyle \cos(39^\circ)\)
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\(\displaystyle \cos(156^\circ)\)
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\(\displaystyle \tan(156^\circ)\)
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None of the above
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3.

Simplify: \(\cos^2(4 x)-\sin^2(4 x)\)

\(\displaystyle \tan(2 x)\)
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\(\displaystyle \cos(2 x)\)
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\(\displaystyle \cos(8 x)\)
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\(\displaystyle \sin(8 x)\)
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\(\displaystyle \sin(2 x)\)
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\(\displaystyle \tan(8 x)\)
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None of the above
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Simplify: \(12 \sin(8 x)\cos(8 x)\)

\(\displaystyle 24 \sin(16 x)\)
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\(\displaystyle 6 \cos(4 x)\)
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\(\displaystyle 6 \cos(16 x)\)
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\(\displaystyle 24 \sin(4 x)\)
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\(\displaystyle 6 \sin(16 x)\)
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\(\displaystyle 6 \sin(4 x)\)
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None of the above
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4.

If \(\displaystyle{\sin(x)=\frac{1}{5}}\) and \(x\) is in quadrant I, find the exact values of the following without solving for \(x\text{:}\)

\(\sin(2x) =\)

\(\cos(2x) =\)

\(\tan(2x) =\)

Answer 1.

\(\frac{2\sqrt{5^{2}-1}}{5^{2}}\)

Answer 2.

\(1-\frac{2}{5^{2}}\)

Answer 3.

\(\frac{2\sqrt{5^{2}-1}}{5^{2}-2}\)

5.

If \(\displaystyle{\cos(x)=-\frac{1}{4}}\) and \(x\) is in quadrant III, find the exact values of the following without solving for \(x\text{:}\)

\(\sin(2x) =\)

\(\cos(2x) =\)

\(\tan(2x) =\)

Answer 1.

\(\frac{2\sqrt{4^{2}-1}}{4^{2}}\)

Answer 2.

\(\frac{2}{4^{2}}-1\)

Answer 3.

\(\frac{2\sqrt{4^{2}-1}}{2-4^{2}}\)

6.

Find the exact value using the half-angle formulas.

\(\displaystyle{ \cos\left(-\frac{5\pi}{8}\right) = }\)

Answer.

\(\frac{-\sqrt{2-\sqrt{2}}}{2}\)

7.

Find the values of the six trigonometric functions evaluated at \(\theta\) if the following conditions hold: \(\displaystyle{\cos(2\theta)=\frac{3}{4}}\) and \(90^\circ \leq \theta \leq 180^\circ\)

\(\sin(\theta) =\)

\(\cos(\theta) =\)

\(\tan(\theta) =\)

\(\sec(\theta) =\)

\(\csc(\theta) =\)

\(\cot(\theta) =\)

Answer 1.

\(\sqrt{\frac{4-3}{2\cdot 4}}\)

Answer 2.

\(-\sqrt{\frac{3+4}{2\cdot 4}}\)

Answer 3.

\(-\sqrt{\frac{4-3}{3+4}}\)

Answer 4.

\(-\sqrt{\frac{2\cdot 4}{3+4}}\)

Answer 5.

\(\sqrt{\frac{2\cdot 4}{4-3}}\)

Answer 6.

\(-\sqrt{\frac{3+4}{4-3}}\)

8.

Find an identity for \(\cos(4t)\) in terms of \(\cos(t)\text{.}\)

\(\cos(4t)\) =

Answer.

\(2\mathopen{}\left(2\mathopen{}\left(\cos\mathopen{}\left(t\right)\right)^{2}-1\right)^{2}-1\)

Solution.

SOLUTION\(\cos(2x)=2\cos^{2}x -1\text{,}\)\(x\)\(2\theta\text{.}\)

\begin{equation*} \begin{aligned} \cos4\theta \amp =\cos(2x) \\ \amp =2\cos^2x-1\quad (\hbox{using the identity for } \cos(2x))\\ \amp =2(2\cos^2\theta-1)^2-1\quad (\hbox{using the identity for } \cos(2\theta)) \end{aligned} \end{equation*}

9.

Use an identity to find the exact value of the expression:

Note: You are not allowed to use decimals in your answer.

\(\displaystyle{\sin\left(\frac{11\pi}{8}\right)}\) =

Answer.

\(-0.923879532511287\)

10.

Use the product to sum formula to fill in the blanks in the identity below:

\(\sin(12 x)\cos(8 x) = \frac{1}{2}(\sin (\) \(x)+\sin(\) \(x))\)

Answer 1.

\(12-8\)

Answer 2.

\(12+8\)

11.

By using known trig identities, \(\frac{\sin(2x)}{1+\cos(2x)}\) can be written as

\(\displaystyle \sec(x)\)
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\(\displaystyle \tan(x)\)
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\(\displaystyle \tan(2x)\)
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\(\displaystyle \csc(2x)\)
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All of the above
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None of the above
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12.

Find \(\cos 2 \theta\) if \(\sin \theta = \frac{5}{13}\text{.}\)

\(\cos 2 \theta =\)

Answer.

\(0.704142\)

Solution.

\(\cos 2 \theta = \cos^2 \theta - \sin^2 \theta\)\(\sin^2 \theta + \cos^2 \theta = 1\text{,}\)\(\cos 2 \theta = 1 - 2 \sin^2 \theta\text{.}\)\(\cos 2 \theta = 1 - 2 (\frac{25}{169}) = 0.704142\text{.}\)

13.

Use the fundamental identities to fully simplify the expression.

`3sin^3(y)csc(y)+cos^2(y)+2cos(-y)cos(y)`

Answer.

\(3\)

14.

Use the fundamental identities to fully simplify the expression.

`(1+tan^2x)/csc^2x+sin^2x+1/(sec^2x)`

Answer.

\(\sec^{2}\mathopen{}\left(x\right)\)

15.

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

`(secx+cscx)/(1+tanx)`, `sin x`

Answer.

\(\frac{1}{\sin\mathopen{}\left(x\right)}\)

16.

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

`1/(sinxcosx)-cotx`, `cot x`

Answer.

\(\frac{1}{\cot\mathopen{}\left(x\right)}\)

17.

Use the fundamental identities to fully simplify the expression.

`tan(-t)sec(-t)cot(-t)`

If the video does not work, click here to watch on YouTube.

Answer.

\(\sec\mathopen{}\left(t\right)\)

18.

Use the fundamental identities to fully simplify the expression.

`cotx cosx + cscx sin^2x`

Answer.

\(\csc\mathopen{}\left(x\right)\)

19.

Use the fundamental identities to fully simplify the expression.

`secx+sin(-x)tanx`

Answer.

\(\cos\mathopen{}\left(x\right)\)

20.

Use the fundamental identities to fully simplify the expression.

`(cott+tant)/(sec(-t))`

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Answer.

\(\csc\mathopen{}\left(t\right)\)