Sin, Cos, Tan - Trigonometry

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Question

If cos x = 0.2 $\small \cosx$ and sin x = 0.4, what is the value of tan x?

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Answer

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

$\small \small $\tan{x}$=\frac{0.4}{0.2}$$

$\small $\tan{x}$=0.035$

$\small $x=\tan^{-1}$0.035$

$\small x=2$

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Question

Find the value of the trigonometric function in fraction form for triangle .

What is the cosine of $\angle B$ ?

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Answer

The cosine of an angle is the value of the adjacent side over the hypotenuse.

Therefore:

$cos \angle B = $\frac{adjacent}{hypotenuse}$ = $\frac{7}{25}$$

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Question

Which of the following describes the ratio of sine?

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Answer

Sine is by definition of sides in a right triangle is opposite side over hypotenuse.

To remember this, use SOH CAH TOA.

SOH: Sine=Opposite/Hypotenuse

CAH: Cosine=Adjacent/Hypotenuse

TOA: Tangent=Opposite/Adjacent

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Question

What is the value of ?

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Answer

Solve each term separately.

$cos(30)= $\frac{\sqrt3}{2}$$

$cos(60)=\frac{1}{2}$$

Add both terms.

$\frac{\sqrt3}{2}$+\frac{1}{2}$= $\frac{\sqrt3+1}{2}$

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Question

Determine the value of .

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Answer

Rewrite in terms of sines and cosines.

$2tan(120)=2($\frac{sin(120)}{cos(120)}$)=2($\frac{\frac{\sqrt3}{2}$}{-$\frac{1}{2}$})$

Simplify the complex fraction.

$2($\frac{\frac{\sqrt3}{2}$}{-$\frac{1}{2}$})= 2\times$\frac{\sqrt3}{2}$\times-2 = -2\sqrt3$

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Question

Find the value of $$\frac{1}{2}$sin(45)+ tan(60)$ .

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Answer

To find the value of $$\frac{1}{2}$sin(45)+ tan(60)$ , solve each term separately.

$\frac{1}{2}$sin(45)=\frac{1}{2}$ \cdot $\frac{\sqrt2}{2}$ = $\frac{\sqrt2}{4}$

$tan(60) = \sqrt3$

Sum the two terms.

$\frac{\sqrt2}{4}$+\sqrt3 = $\frac{\sqrt2}{4}$+\frac{4\sqrt3}{4}$ = $\frac{\sqrt2+4\sqrt3}{4}$

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Question

Select the ratio that would give Tan B.

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Answer

We need the Tan B. Which side lengths correspond to this ratio?

$\tan=\frac{opp}{adj}$$

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Question

Calculate $tan($\frac{4\pi}{3}$)$ .

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Answer

The tangent function has a period of $\pi$ units. That is,

$tan(x+n\pi)=tanx$

for all $n\in\mathbb{Z}$ .

Since $$\frac{4\pi}{3}$=\frac{\pi}{3}$+\pi$ , we can rewrite the original expression $tan($\frac{4\pi}{3}$)$ as follows:

$tan($\frac{4\pi}{3}$)=tan($\frac{\pi}{3}$+\pi)$

$=tan($\frac{\pi}{3}$)$

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

$=\frac{(\frac{\sqrt{3}$}{2})}{($\frac{1}{2}$)}$

$=$\sqrt{3}$$

Hence,

$tan($\frac{4\pi}{3}$)=$\sqrt{3}$$

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Question

Calculate $cos($\frac{5\pi}{3}$)$ .

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Answer

First, convert the given angle measure from radians to degrees:

Next, recall that lies in the fourth quadrant of the unit circle, wherein the cosine is positive. Furthermore, the reference angle of is

Hence, all that is required is to recognize from these observations that

,

which is $\frac{1}{2}$ .

Therefore,

$cos($\frac{5\pi}{3}$)=\frac{1}{2}$$

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Question

What is the result when the following expression is simplified as much as possible?

$\sin(2h)\sec(h)+2\sin(-h)$

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Answer

Because $\sin x$ is an odd function, we can rewrite the second term in the expression.

$2\sin(-h)=-2\sin h$ .

We now use a double-angle formula to expand the first term.

$\sin(2h)\sec h=2\sin h\cos h\sec h$ .

Because they are reciprocals, $\cos h\sec h=1$ .

$2\sin h\cos h\sec h-2\sin h=2\sin h-2\sin h=0.$

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Question

Round to the nearest hundredth.

Use your calculator to find:

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Answer

Before plugging the function into the calculator make sure the mode of the calculator is set to degrees,

Plug in which equals to .

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Question

Round to the nearest hundredth.

Use your calculator to find:

$\cos $\frac{3\pi}{8}$$

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Answer

Before plugging the function into the calculator make sure the mode of the calculator is set to radians,

plug in

$\cos $\frac{3\pi}{8}$$

which is equal to .

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Question

$\left ( \sin15 + \cos\15 \right $)^{2}$$ What is the closest value to this expression?

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Answer

First you have to know the equation for $\left ( a+b\right $)^{2}$ = $a^{2}$$+2ab+b^{2}$$ and apply it to our equation. Second the formula of will allow you to get rid of the square sin and cos. Third the equation $\sin2x = 2\sin \cos$ will allow you to get rid of the . Therefore, we will have a $\sin(30)$ , and that will be equal to . Sum and and you will get

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Question

if $\cos x = $\frac{4}{5}$$ What is $\sin2x$ ?

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Answer

Remember two things. First if $\cos x=\frac{4}{5}$$ , find the $\sin(x)$ by using the Pythagoras Theorem. If one side is and the hypotenuse is , then the other side is . $\sin x$ will be $\frac{3}{5}$ . Finally remember the formula for $\sin2x = 2\sin x *\cos x$ . And just place the things we found to the equation.

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Question

What is the value of cos30o . sin30o . tan60o

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Answer

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Question

If cos x = 0.2 $\small \cosx$ and sin x = 0.4, what is the value of tan x?

Tap to reveal answer

Answer

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

$\small \small $\tan{x}$=\frac{0.4}{0.2}$$

$\small $\tan{x}$=0.035$

$\small $x=\tan^{-1}$0.035$

$\small x=2$

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Question

Find the value of the trigonometric function in fraction form for triangle .

What is the cosine of $\angle B$ ?

Tap to reveal answer

Answer

The cosine of an angle is the value of the adjacent side over the hypotenuse.

Therefore:

$cos \angle B = $\frac{adjacent}{hypotenuse}$ = $\frac{7}{25}$$

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Question

Which of the following describes the ratio of sine?

Tap to reveal answer

Answer

Sine is by definition of sides in a right triangle is opposite side over hypotenuse.

To remember this, use SOH CAH TOA.

SOH: Sine=Opposite/Hypotenuse

CAH: Cosine=Adjacent/Hypotenuse

TOA: Tangent=Opposite/Adjacent

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Question

What is the value of ?

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Answer

Solve each term separately.

$cos(30)= $\frac{\sqrt3}{2}$$

$cos(60)=\frac{1}{2}$$

Add both terms.

$\frac{\sqrt3}{2}$+\frac{1}{2}$= $\frac{\sqrt3+1}{2}$

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Question

Determine the value of .

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Answer

Rewrite in terms of sines and cosines.

$2tan(120)=2($\frac{sin(120)}{cos(120)}$)=2($\frac{\frac{\sqrt3}{2}$}{-$\frac{1}{2}$})$

Simplify the complex fraction.

$2($\frac{\frac{\sqrt3}{2}$}{-$\frac{1}{2}$})= 2\times$\frac{\sqrt3}{2}$\times-2 = -2\sqrt3$

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