Understanding Sine, Cosine, and Tangent - Math

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Question

What is the $\sin C$ ?

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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?

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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?

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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 $30^\circ$ ?

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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 $2\pi$ ?

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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

What is the $\sin C$ ?

Tap to reveal answer

Answer

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

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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.

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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 $(\sqrt2,-\sqrt2)$ .

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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}$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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}$}{2}$ .

← Didn't Know|Knew It →

Question

What is the $\sin C$ ?

Tap to reveal answer

Answer

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

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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.

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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 $(\sqrt2,-\sqrt2)$ .

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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}$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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}$}{2}$ .

← Didn't Know|Knew It →

Question

What is the $\sin C$ ?

Tap to reveal answer

Answer

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

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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.

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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 $(\sqrt2,-\sqrt2)$ .

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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}$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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}$}{2}$ .

← Didn't Know|Knew It →

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