Q: A rain gutter is constructed from a 12-inch wide aluminum sheet. After marking off 4 inches from each edge, the sides are bent up at an angle $$\theta. $$What is the area $$A of $$the opening as a function of $$\theta$$?

$$A(\theta) = 16 \sin(\theta) + 16 \sin(\theta) \cos(\theta)$$

Q: If the angle of elevation from a person to an object is $$\theta$$, which trigonometric function relates the height of the object to the horizontal distance?

$$\tan(\theta) = \frac{\text{height}}{\text{distance}}$$

Q: In a right triangle, if $$\theta is an $$acute angle, which of the following is the correct expression for $$\cot \theta$$?

$$\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$$

Q: A triangle has sides $$a = 6$$, $$b = 8$$, and angle $$A = 30^\circ. $$What is the area of the triangle?

$$\text{Area} = \frac{1}{2} \times 6 \times 8 \times \sin(30^\circ)$$

Q: Given a triangle with sides $$a = 4$$, $$b = 2$$, and angle $$C = 45^\circ$$, what is the length of side $$c$$ opposite angle $$C$$?

$$c = \sqrt{4^2$ + 2^2$ - 2 \times 4 \times 2 \times \cos(45^\circ)}$$

Q: If the sides of a triangle are $$a = 3$$, $$b = 2$$, and the included angle $$C = 60^\circ$$, what is the area of the triangle?

$$\text{Area} = \frac{1}{2} \times 3 \times 2 \times \sin(60^\circ)$$

Q: If the sides of a triangle are $$a = 6$$, $$b = 8$$, and the included angle $$A = 30^\circ$$, what is the area of the triangle?

$$\text{Area} = \frac{1}{2} \times 6 \times 8 \times \sin(30^\circ)$$

Question 1

Given a right triangle with sides labeled as $$a ($$adjacent), $$o ($$opposite), and $$h ($$hypotenuse), which of the following correctly expresses $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = \frac{o}{h}$$

Question 2

A rain gutter is constructed from a 12-inch wide aluminum sheet. After marking off 4 inches from each edge, the sides are bent up at an angle $$	heta. $$What is the area $$A of $$the opening as a function of $$	heta$$?

Accepted Answer

$$A(	heta) = 16 \sin(	heta) + 16 \sin(	heta) \cos(	heta)$$

Question 3

If $$	riangle ABC is a $$right triangle with $$A + B = 90^\circ$$, which of the following is true about the trigonometric functions of complementary angles?

Accepted Answer

$$\sin A = \cos B$$

Question 4

A person stands 200 meters from a riverbank and measures the angle of elevation to the top of a tree across the river as $$20^\circ. $$What is the height $$b of $$the tree?

Accepted Answer

$$b = 200 	an(20^\circ)$$

Question 5

In the context of measuring cloud height using a light detector and projector, if the base $$b is $$known and the angle $$	heta is $$measured, what formula gives the cloud height $$h$$?

Accepted Answer

$$h = b 	an(	heta)$$

Question 6

Given a right triangle with sides $$a = 3$$, $$b = 2$$, and hypotenuse $$c$$, what is the value of $$c$$?

Accepted Answer

$$c = \sqrt{3^2$ + 2^2$}$$

Question 7

A triangle has angles $$40^\circ$$ and $$60^\circ$$ and a side opposite the $$60^\circ$$ angle of length 4. What is the length of the side opposite the $$40^\circ$$ angle?

Accepted Answer

$$b = \frac{4 \sin(40^\circ)}{\sin(60^\circ)}$$

Question 8

If the angle of elevation from a person to an object is $$	heta$$, which trigonometric function relates the height of the object to the horizontal distance?

Accepted Answer

$$	an(	heta) = \frac{	ext{height}}{	ext{distance}}$$

Question 9

In a right triangle, if $$	heta is an $$acute angle, which of the following is the correct expression for $$\cot 	heta$$?

Accepted Answer

$$\cot 	heta = \frac{	ext{adjacent}}{	ext{opposite}}$$

Question 10

A triangle has sides $$a = 6$$, $$b = 8$$, and angle $$A = 30^\circ. $$What is the area of the triangle?

Accepted Answer

$$	ext{Area} = \frac{1}{2} 	imes 6 	imes 8 	imes \sin(30^\circ)$$

Question 11

If $$\sin B = \cos A in a $$right triangle, what is the relationship between angles $$A$$ and $$B$$?

Accepted Answer

$$A + B = 90^\circ$$

Question 12

A light beam is projected vertically to the base of clouds. If the base is $$b$$ meters and the angle of elevation is $$	heta$$, which formula gives the height $$h of $$the clouds?

Accepted Answer

$$h = b 	an(	heta)$$

Question 13

Given a triangle with sides $$a = 4$$, $$b = 2$$, and angle $$C = 45^\circ$$, what is the length of side $$c$$ opposite angle $$C$$?

Accepted Answer

$$c = \sqrt{4^2$ + 2^2$ - 2 	imes 4 	imes 2 	imes \cos(45^\circ)}$$

Question 14

If the sides of a triangle are $$a = 3$$, $$b = 2$$, and the included angle $$C = 60^\circ$$, what is the area of the triangle?

Accepted Answer

$$	ext{Area} = \frac{1}{2} 	imes 3 	imes 2 	imes \sin(60^\circ)$$

Question 15

A triangle has sides $$a = 5$$, angle $$B = 35^\circ$$, and angle $$A = 15^\circ. $$What is the length of side $$b$$ opposite angle $$B$$?

Accepted Answer

Question 16

If the sides of a triangle are $$a = 6$$, $$b = 8$$, and the included angle $$A = 30^\circ$$, what is the area of the triangle?

Accepted Answer

$$	ext{Area} = \frac{1}{2} 	imes 6 	imes 8 	imes \sin(30^\circ)$$

Right Triangle Trigonometry and Applications

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