Trigonometric Operations

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Trig_id

What is $\theta$ if and ?

$51.34^{\circ}$

$53.14^{\circ}$

$55.14^{\circ}$

$0.90^{\circ}$

$89.60^{\circ}$

Explanation

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.

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

$\sec{x}$

$\csc{x}$

$\sin x$

$\cot{x}$

$\cos{x}$

Explanation

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

Trig_id

What is $\theta$ if and ?

$51.34^{\circ}$

$53.14^{\circ}$

$55.14^{\circ}$

$0.90^{\circ}$

$89.60^{\circ}$

Explanation

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

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

$\tan=1.25$

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.

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

$\sec{x}$

$\csc{x}$

$\sin x$

$\cot{x}$

$\cos{x}$

Explanation

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

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

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

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

$\frac{1}{2}$

$\frac{1}{3}$

$\sqrt{3}$

Explanation

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.

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

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

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

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

$\frac{1}{2}$

$\frac{1}{3}$

$\sqrt{3}$

Explanation

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.

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

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

$\frac{5}{3}$

$\frac{5}{4}$

$\frac{3}{5}$

$\frac{3}{4}$

$\frac{4}{3}$

Explanation

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.

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

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

$\frac{5}{3}$

$\frac{5}{4}$

$\frac{3}{5}$

$\frac{3}{4}$

$\frac{4}{3}$

Explanation

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.

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

Triangle

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

$x\cos\theta$

$x\sin\theta$

$x\tan\theta$

$\cos(x\times \theta)$

$\frac{\sin(x)}{\theta}$

Explanation

$\cos\theta$ 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.

Triangle

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

$x\cos\theta$

$x\sin\theta$

$x\tan\theta$

$\cos(x\times \theta)$

$\frac{\sin(x)}{\theta}$

Explanation

$\cos\theta$ 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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