Problem Statement

rand(2,8)
5+1
round(sqrt(6*6+5*5),2)
round(5/7.81,2)
round(6/7.81,2)
round(7.81/6,2)
round(6/5,2)

Assume that . Find sin(θ), sec(θ), and cot(θ) if tan(θ)=.

 
Step 1

Recall the signs of sin(θ) and cos(θ) for .

Question Sequence

Question 1

For , sin(θ) and cos(θ) are . Thus we can use the definitions of the trigonometric functions associated with a right triangle to find sin(θ), sec(θ), and cot(θ).

Incorrect
Correct

 
Step 2

Draw a right triangle to find sin(θ) and cos(θ) given that tan(θ)=.

Label θ as one of the acute angles inside the right triangle. Label the leg of the triangle opposite θ as b and label the leg of the triangle adjacent to θ as a. Label the hypotenuse c.

According to this triangle, we can solve for b and a using the relationship

.

Question Sequence

Question 2

Since , we have b = 5 and a = .

Incorrect
Correct

Question 3

We can find c using the Pythagorean theorem, which states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

a2 + b2 = c2

c =

(Rounded to two decimal places.)

Incorrect
Correct

 
Step 3

Knowing that a = 6, b = 5, and c = 7.81, we can solve for sin(θ) and cos(θ) using their definitions with respect to a right triangle.

Question Sequence

Question 4

= .

2
Correct.
Incorrect.

Question 5

=.

Correct.
Incorrect.

Question 6

For and tan(θ)=,

sin(θ) =

and

cos(θ) = .

(Rounded to two decimal places.)

Correct.
Incorrect.

 
Step 4

Recall the definitions of sec(θ) and cot(θ).

Question 7

sec(θ) =

cot(θ) =

Correct.
Incorrect.

 
Step 5

Find and simplify sec(θ) and cot(θ) using, from Step 3, sin(θ) = 0.64 and cos(θ) = 0.77.

Question Sequence

Question 8

For and tan(θ) = we have

sec(θ) =

(Rounded to two decimal places.)

Correct.
Incorrect.

Question 9

For and tan(θ) = we have

cot(θ) =

(Rounded to two decimal places.)

Correct.
Incorrect.