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High School Math : Trigonometric Identities

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Example Question #1 : Using Basic And Definitional Identities

Trig_id

What is the $\tan$ of $\theta$ ?

Possible Answers:

$\frac{a}{h}$

$\frac{o}{a}$

$\frac{a}{o}$

$\frac{o}{h}$

$\frac{a}{h}$

Correct answer:

$\frac{o}{a}$

Explanation:

When working with basic trigonometric identities, it's easiest to remember the mnemonic: SOHCAHTOA .

$\sin=\frac{opposite}{hypotenuse}$

$\cos=\frac{adjacent}{hypotenuse}$

$\tan=\frac{opposite}{adjacent}$

When one names the right triangle, the opposite side is opposite to the angle, the adjacent side is next to the angle, and the hypotenuse spans the two legs of the right angle.

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Example Question #2 : Using Basic And Definitional Identities

Simplify $(1-\cos^2\theta )\cdot\cot\theta\cdot\csc^2\theta$ .

Possible Answers:

$\tan\theta$

$\sin^2\theta$

$\sin\theta$

$\cot\theta$

$\cos^2\theta$

Correct answer:

$\cot\theta$

Explanation:

Simplifying trionometric expressions or identities often involves a little trial and error, so it's hard to come up with a strategy that works every time. A lot of times you have to try multiple strategies and see which one helps.

Often, if you have any form of $\tan\theta,$ $\cot\theta,$ $\csc\theta,$ or $\sec\theta,$ in an expression, it helps to rewrite it in terms of sine and cosine. In this problem, we can use the identities $\cot\theta=\frac{\cos\theta }{\sin\theta }$ and $\csc\theta=\frac{1}{\sin\theta }$ .

$(1-\cos^2\theta )\cdot\cot\theta\cdot\csc^2\theta=(1-\cos^2\theta )\cdot\frac{\cos\theta }{\sin\theta }\cdot(\frac{1}{\sin\theta})^2$

$=(1-\cos^2\theta )\cdot\frac{\cos\theta }{\sin\theta }\cdot\frac{1}{\sin^2\theta}$ .

This doesn't seem to help a whole lot. However, we should recognize that $(1-\cos^2\theta )=\sin^2\theta$ because of the Pythagorean identity $\sin^2\theta+\cos^2\theta=1$ .

$(1-\cos^2\theta )\cdot\frac{\cos\theta }{\sin\theta }\cdot\frac{1}{\sin^2\theta}=(\sin^2\theta )\cdot\frac{\cos\theta }{\sin\theta }\cdot\frac{1}{\sin^2\theta}$

We can cancel the $\sin^2\theta$ terms in the numerator and denominator.

$(\sin^2\theta )\cdot\frac{\cos\theta }{\sin\theta }\cdot\frac{1}{\sin^2\theta}=\frac{\cos\theta }{\sin\theta }=\cot\theta$ .

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Example Question #111 : Trigonometry

Trig_id

What is the $\sin$ of $\theta$ ?

Possible Answers:

$\frac{h}{a}$

$\frac{a}{h}$

$\frac{h}{o}$

$\frac{o}{h}$

$\frac{o}{a}$

Correct answer:

$\frac{o}{h}$

Explanation:

When working with basic trigonometric identities, it's easiest to remember the mnemonic: SOHCAHTOA .

$\sin=\frac{opposite}{hypotenuse}$

$\cos=\frac{adjacent}{hypotenuse}$ ,

$\tan=\frac{opposite}{adjacent}$

When one names the right triangle, the opposite side is opposite to the angle, the adjacent side is next to the angle, and the hypotenuse spans the two legs of the right angle.

Report an Error

Example Question #4 : Using Basic And Definitional Identities

Trig_id

What is the $\cos$ of $\theta$ ?

Possible Answers:

$\frac{h}{o}$

$\frac{o}{h}$

$\frac{a}{h}$

$\frac{o}{a}$

$\frac{h}{a}$

Correct answer:

$\frac{a}{h}$

Explanation:

When working with basic trigonometric identities, it's easiest to remember the mnemonic: SOHCAHTOA .

$\sin=\frac{opposite}{hypotenuse}$

$\cos=\frac{adjacent}{hypotenuse}$ ,

$\tan=\frac{opposite}{adjacent}$

When one names the right triangle, the opposite side is opposite to the angle, the adjacent side is next to the angle, and the hypotenuse spans the two legs of the right angle.

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Example Question #11 : Relations And Functions

Simplify

$\cot(x)\times{\sin(x) }$

Possible Answers:

$\arcsin(x)$

$\csc(x)$

$\arcsin(\pi )$

$\cos(x)$

$\tan(x)$

Correct answer:

$\cos(x)$

Explanation:

$\cot(x) = \cos(x)/\sin(x)$ . Thus: $[\cos(x)/\sin(x)]\times\sin(x)=\cos(x)$

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Example Question #1 : Trigonometric Identities

Simplify

$\frac{cot (x)\cdot tan(x)}{cos(x)}$

Possible Answers:

$\cot(x)/sec(x)$

$\csc(x)$

$\cos(x)\times\tan(x)$

$\sec(x)$

$\cos(x)$

Correct answer:

$\sec(x)$

Explanation:

$\cot(x)\times\tan(x) = 1$

and

$1/(\cos(x)) = \sec(x)$ .

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Example Question #3 : Using Pythagorean Identities

Simplify $\frac{\cos^2x-1}{\sin x}$ .

Possible Answers:

$\tan x$

$\sin^2 x$

$\sin x$

$\sin x\cos x$

$-\sin x$

Correct answer:

$-\sin x$

Explanation:

Remember that $\sin^2x+\cos^2 x=1$ . We can rearrange this to simplify our given equation:

$\frac{\cos^2x-1}{\sin x}$

$=\frac{-\sin^2x}{\sin x}=-\sin x$

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Example Question #4 : Using Pythagorean Identities

Simplify:

$\sin^2x+\cos^2x+\tan^2x$

Possible Answers:

$\sin x\cos x$

$\csc^2 x$

$\sec^2x$

$2\cos x+\cot x$

This is the most simplified version.

Correct answer:

$\sec^2x$

Explanation:

Whenever you see a trigonometric function squared, start looking for a Pythagorean identity.

The two identities used in this problem are $\sin^2 x + \cos ^2 x=1$ and $\tan^2 x +1 =\sec^2 x$ .

Substitute and solve.

$\sin^2x+\cos^2x+\tan^2x=?$

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

$(1)+\tan^2x=\sec^2 x$

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Example Question #1 : Using Identities Of Squared Functions

Factor and simplify $\frac{\sin^2x-9}{\sin x-3}$ .

Possible Answers:

$\csc x$

$\cos^2 x+3$

$\sin x+3$

$\sin x \cos x$

This is already it's most reduced form.

Correct answer:

$\sin x+3$

Explanation:

To reduce $\frac{\sin^2x-9}{\sin x-3}$ , factor the numerator: $\frac{(\sin x-3)(\sin x+3)}{\sin x-3}$

Notice that we can cancel out a $\sin x -3$ .

This leaves us with $\sin x +3$ .

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Example Question #1 : Using Sum And Product Identities

Simplify $\sin x\cot x$ .

Possible Answers:

$\csc x$

$\sin x \cos x$

$\tan x$

$cos \, x$

$\cos^2x$

Correct answer:

$cos \, x$

Explanation:

To simplify $\sin x\cot x$ , break them into their SOHCAHTOA parts:

$\frac{\text{opposite}}{\text{hypotenuse}}*\frac{\text{adjacent}}{\text{opposite}}$ .

Notice that the opposite's cancel out, leaving $\frac{\text{adjacent}}{\text{hypotenuse}}=\cos x$ .

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High School Math : Trigonometric Identities

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Using Basic And Definitional Identities

Example Question #2 : Using Basic And Definitional Identities

Example Question #111 : Trigonometry

Example Question #4 : Using Basic And Definitional Identities

Example Question #11 : Relations And Functions

Example Question #1 : Trigonometric Identities

Example Question #3 : Using Pythagorean Identities

Example Question #4 : Using Pythagorean Identities

Example Question #1 : Using Identities Of Squared Functions

Example Question #1 : Using Sum And Product Identities

All High School Math Resources

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