Question

Consider the right triangle shown below.

A diagram of a right triangle with one interior angle measure labeled theta radians. The side opposite the angle is labeled y, the side adjacent to the angle is labeled x, and the hypotenuse is labeled r.

Use the Pythagorean Theorem to write an equation relating
x
,
y
, and
r
.

x^2+y^2=r^2
Correct

Write a formula that expresses
x
in terms of
r
and
θ
.

x
=

r cos
Incorrect

Write a formula that expresses
y
in terms of
r
and
θ
.

y
=

r sin(0)
Incorrect

Use your answers to parts (a)-(c) to write an equation relating
cos
(
θ
)
,
sin
(
θ
)
, and
r
.

12
Incorrect

Answer 1

Let's go through each part one by one:

(a) The equation relating x, y, and r is indeed x^2 + y^2 = r^2. This is known as the Pythagorean Theorem and it applies to right triangles.

(b) To express x in terms of r and θ, we use the cosine function. The correct formula is x = r * cos(θ). The cosine function relates the adjacent side (x) to the hypotenuse (r) and the angle (θ) in a right triangle.

(c) Similarly, to express y in terms of r and θ, we use the sine function. The correct formula is y = r * sin(θ). The sine function relates the opposite side (y) to the hypotenuse (r) and the angle (θ) in a right triangle.

(d) Now, to relate cos(θ), sin(θ), and r, we can use the Pythagorean Identity. It states that cos^2(θ) + sin^2(θ) = 1. By rearranging the equation, we can write it as r^2 = cos^2(θ) + sin^2(θ).

So, the equation relating cos(θ), sin(θ), and r is r^2 = cos^2(θ) + sin^2(θ).

I hope this clears things up! If you have any more questions or need further assistance, feel free to ask!