Trigonometry - Math

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Question

What is $\theta$ if and ?

Answer

In order to find $\theta$ we need to utilize the given information in the problem. We are given the opposite and adjacent sides. We can then, by definition, find the $\tan$ of $\theta$ and its measure in degrees by utilizing the $\arctan$ function.

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

$\tan=\frac{10}{8}$

$\tan=1.25$

Now to find the measure of the angle using the $\arctan$ function.

$\theta=\arctan1.25$

$\rightarrow 51.34^{\circ}$

If you calculated the angle's measure to be $0.90^{\circ}}$ then your calculator was set to radians and needs to be set on degrees.

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Question

$o=7.6, h=15.7, \Theta = ?$

Answer

In order to find $\theta$ we need to utilize the given information in the problem. We are given the opposite and hypotenuse sides. We can then, by definition, find the $\sin$ of $\theta$ and its measure in degrees by utilizing the $\arcsin$ function.

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

$\sin=\frac{7.6}{15.7}$

$\sin=0.4841$

Now to find the measure of the angle using the $\arcsin$ function.

$\theta=\arcsin0.4841$

$\rightarrow 28.95^{\circ}$

If you calculated the angle's measure to be $\dpi{100} 0.50^{\circ}}$ then your calculator was set to radians and needs to be set on degrees.

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Question

$\sin(x^\circ)=\frac{1}{2}$

What is ?

Answer

To get rid of $\sin(x^\circ)$ , we take the $\arcsin$ or $\sin^{-1}$ of both sides.

$\sin(x^\circ)=\frac{1}{2}$

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

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

$x^\circ=30^\circ$

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Question

An angle has a cosine of $\frac{4}{5}$ . What will its cosecant be?

Answer

The problem tells us that the cosine of the angle will be $\frac{4}{5}$ . Cosine is the adjacent over the hypotenuse. From here we can use the Pythaogrean theorem:

$\text{Opposite}^2+\text{Adjacent}^2=\text{Hypotenuse}^2$

$\sqrt{O^2}=\sqrt{9}$

Now we know our opposite, adjacent, and hypotenuse.

The cosecant is $\csc=\frac{\text{Hypotenuse}}{\text{Opposite}}$ .

From here we can plug in our given values.

$\csc=\frac{5}{3}$

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Question

Which of these is equal to $\frac{\text{hypotenuse}}{\text{adjacent}}$ for angle ?

Answer

$\sec{x}=\frac{\text{hypotenuse}}{\text{adjacent}}$ , as it is the inverse of the $\cos{x}$ function. This is therefore the answer.

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Question

What is the $\sin C$ ?

Answer

$\sin X = \frac{OPPOSITE}{HYPOTENUSE}$

$\sin C = \frac{AB}{AC}$

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Question

In the right triangle above, which of the following expressions gives the length of y?

Answer

$\cos\theta$ is defined as the ratio of the adjacent side to the hypotenuse, or in this case $\frac{y}{x}$ . Solving for y gives the correct expression.

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Question

If the polar coordinates of a point are $\left (2,\frac{-\pi}{4} \right )$ , then what are its rectangular coordinates?

Answer

The polar coordinates of a point are given as $(r,\theta )$ , where r represents the distance from the point to the origin, and $\theta$ represents the angle of rotation. (A negative angle of rotation denotes a clockwise rotation, while a positive angle denotes a counterclockwise rotation.)

The following formulas are used for conversion from polar coordinates to rectangular (x, y) coordinates.

$x=r\cos\theta$

$y=r\sin\theta$

In this problem, the polar coordinates of the point are $\left (2,\frac{-\pi}{4} \right )$ , which means that and $\theta=-\frac{\pi}{4}$ . We can apply the conversion formulas to find the values of x and y.

$x=r\cos\theta=2\cos\left (\frac{-\pi}{4} \right )=2\cdot \frac{\sqrt2}{2}= \sqrt{2}$

$y=r\sin\theta=2\sin\left (\frac{-\pi}{4} \right )=2\cdot \frac{-\sqrt2}{2}= -\sqrt{2}$

The rectangular coordinates are $(\sqrt2,-\sqrt2)$ .

The answer is $(\sqrt2,-\sqrt2)$ .

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Question

What is the cosine of $30^\circ$ ?

Answer

The pattern for the side of a triangle is $x:x\sqrt{3}:2x$ .

Since $\cos=\frac{\text{adjacent}}{\text{hypotenuse}}$ , we can plug in our given values.

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

$\cos=\frac{x\sqrt{3}}{2x}$

Notice that the 's cancel out.

$\cos=\frac{\sqrt{3}}{2}$

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Question

If , what is if is between and $2\pi$ ?

Answer

Recall that $sin(\frac{\pi}{6}) =0.5$ .

Therefore, we are looking for $sin(8*\frac{\pi}{6})$ or $sin(\frac{4\pi}{3})$ .

Now, this has a reference angle of $\frac{\pi}{3}$ , but it is in the third quadrant. This means that the value will be negative. The value of $sin(\frac{\pi}{3})$ is $\frac{\sqrt{3}}{2}$ . However, given the quadrant of our angle, it will be $-\frac{\sqrt{3}}{2}$ .

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Question

Solve for and .

(Figure not drawn to scale).

Answer

The angles containing the variable all reside along one line, therefore, their sum must be .

$\small 5x+4x+3x=180^o$

$\small 12x=180^o$

$\small x=15^o$

Because and are opposite angles, they must be equal.

$\small 2y=5x$

$\small x=15^o$

$\small 2y=5(15^o)=75^o$

$\small y=\frac{75^o}{2}=37.5^o$

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Question

Which one of these is positive in quadrant III?

Answer

The pattern for positive functions is All Student Take Calculus. In quandrant I, all trigonometric functions are positive. In quadrant II, sine is positive. In qudrant III, tangent is positive. In quadrant IV, cosine is positive.

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Question

Solve for .

(Figure not drawn to scale).

Answer

The angles are supplementary, therefore, the sum of the angles must equal .

$\small (4n+22^o)+(8n-10^o)=180^o$

$\small 4n+22^o+8n-10^o=180^o$

$\small 12n+12^o=180^o$

$\small 12n=168^o$

$\small n=14^o$

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Question

Are $129^{\circ}$ and $51^{\circ}$ supplementary angles?

Answer

Since supplementary angles must add up to $180^{\circ}$ , the given angles are indeed supplementary.

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Question

Are $73^{\circ}$ and $17^{\circ}$ complementary angles?

Answer

Complementary angles add up to $90^{\circ}$ . Therefore, these angles are complementary.

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Question

Which of the following angles is supplementary to $15^\circ$ ?

Answer

When two angles are supplementary, they add up to $180^\circ$ .

For this problem, we can set up an equation and solve for the supplementary angle:

$x^\circ+15^\circ=180^\circ$

$x^\circ=180^\circ-15^\circ$

$x^\circ=165^\circ$

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Question

What angle is complementary to $34^\circ$ ?

Answer

Two complementary angles add up to $90^\circ$ .

Therefore, $34^\circ+x^\circ=90^\circ$ .

$x^\circ=90^\circ-34^\circ$

$x^\circ=56^\circ$

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Question

What angle is supplementary to $131^\circ$ ?

Answer

Supplementary angles add up to $180^\circ$ . That means:

$x+131^\circ=180^\circ$

$x=180^\circ-131^\circ$

$x=49^\circ$

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Question

Which of the following angles is complementary to $32^\circ$ ?

Answer

Two complementary angles add up to $90^\circ$ .

$90^\circ=32^\circ+x^\circ$

$90^\circ-32^\circ=x^\circ$

$58^\circ=x^\circ$

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Question

What angle is supplementary to $142^\circ$ ?

Answer

When two angles are supplementary, they add up to $180^\circ$ .

$142^\circ+x=180^\circ$

Solve for :

$142^\circ+x=180^\circ$

$x=180^\circ-142^\circ$

$x=38^\circ$

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