Question

1. consider the point shown below, which is located on the terminal ray of an angle located in quadrant i of the graph. hint: what is the length of the radius of the circle? (a) use the given information to compute the value of \\( \sin (\theta) \\) (b) use the given information to compute the value of \\( \cos (\theta) \\) (c) use the given information to compute the value of \\( \tan (\theta) \\)

Explanation:

Part (a): Radius Calculation

Step1: Identify coordinates and radius formula

The point on the terminal ray has \( x = 3.8 \, \text{cm} \) (red) and \( y = 2.31 \, \text{cm} \) (blue)? Wait, no, looking at the diagram: the horizontal (x) component is 3.8 cm (red), vertical (y) is 2.31 cm (blue)? Wait, no, the green is 2.8? Wait, no, the right triangle has legs: let's check the diagram. The point has \( x = 3.8 \, \text{cm} \) (red, horizontal), \( y = 2.31 \, \text{cm} \) (blue, vertical)? Wait, no, the angle \( \theta \) is with the x-axis? Wait, the radius \( r \) is the hypotenuse of the right triangle with legs \( x = 3.8 \, \text{cm} \) (adjacent) and \( y = 2.31 \, \text{cm} \) (opposite)? Wait, no, looking at the diagram: the red arrow is 3.8 cm (x-axis, horizontal), blue is 2.31 cm (vertical, y-axis), and the black line is the radius. Wait, maybe the legs are \( x = 3.8 \, \text{cm} \) (horizontal) and \( y = 2.31 \, \text{cm} \) (vertical)? Wait, no, the green is 2.8? Wait, maybe I misread. Wait, the problem says "the point shown below, which is located on the terminal ray". The right triangle has legs: let's use the Pythagorean theorem. So \( r = \sqrt{x^2 + y^2} \), where \( x = 3.8 \, \text{cm} \), \( y = 2.31 \, \text{cm} \)? Wait, no, the diagram has red 3.8 cm (x), blue 2.31 cm (y), and the radius is the hypotenuse. Wait, let's calculate:

Step1: Apply Pythagorean theorem

The radius \( r \) of the circle (distance from origin to the point) is given by \( r = \sqrt{x^2 + y^2} \), where \( x = 3.8 \, \text{cm} \) and \( y = 2.31 \, \text{cm} \) (from the diagram: red is x=3.8, blue is y=2.31). Wait, but maybe the green is 2.8? Wait, no, the problem's hint is "What is the length of the radius of the circle?". Let's check the numbers: 3.8 cm (x), 2.31 cm (y). So:

\( r = \sqrt{(3.8)^2 + (2.31)^2} \)

Calculate \( 3.8^2 = 14.44 \), \( 2.31^2 = 5.3361 \)

Sum: \( 14.44 + 5.3361 = 19.7761 \)

Then \( r = \sqrt{19.7761} \approx 4.447 \, \text{cm} \). Wait, but maybe the legs are 2.8 and 3.8? Wait, the green is 2.8, red is 3.8. Wait, maybe I misread the y-component. Wait, the diagram: blue is 2.31, red is 3.8, green is 2.8. Wait, no, the point is on the terminal ray, so the coordinates are (x, y) = (3.8, 2.31)? Or (3.8, 2.8)? Wait, the green line is 2.8 cm (vertical?), red is 3.8 cm (horizontal), blue is 2.31 cm (vertical?). Wait, maybe the right triangle has legs 3.8 cm (x) and 2.31 cm (y), so radius is hypotenuse. Let's proceed with x=3.8, y=2.31.

Part (a) Solution:

Step1: Identify x and y

From the diagram, the point on the terminal ray has \( x = 3.8 \, \text{cm} \) (horizontal) and \( y = 2.31 \, \text{cm} \) (vertical). The radius \( r \) is the distance from the origin to this point, given by the Pythagorean theorem: \( r = \sqrt{x^2 + y^2} \).

Step2: Calculate \( x^2 \) and \( y^2 \)

\( x^2 = (3.8)^2 = 14.44 \)
\( y^2 = (2.31)^2 = 5.3361 \)

Step3: Sum and take square root

Sum: \( 14.44 + 5.3361 = 19.7761 \)
\( r = \sqrt{19.7761} \approx 4.447 \, \text{cm} \) (or exactly, since \( 19.7761 = 4.447^2 \)? Wait, \( 4.447^2 \approx 19.77 \), yes. Alternatively, maybe the legs are 3.8 and 2.8? Wait, 3.8 and 2.8: \( 3.8^2 + 2.8^2 = 14.44 + 7.84 = 22.28 \), \( \sqrt{22.28} \approx 4.72 \). But the blue is 2.31. Maybe the diagram has x=3.8, y=2.31. Let's confirm with the hint.

Part (b): \( \cos(\theta) \)

Step1: Recall cosine definition

For an angle \( \theta \) in standard position, \( \cos(\theta) = \frac{x}{r} \), where \( x \) is the adjacent side (x-coordinate) and \( r \) is the radius (hypotenuse).

Step2: Substitute values

From part (a), \( r \approx 4.447 \, \text{cm} \), \( x = 3.8 \, \text{cm} \).
\( \cos(\theta) = \frac{3.8}{4.447} \approx 0.8545 \)

Part (c): \( \tan(\theta) \)

Answer:

Step1: Recall tangent definition

For an angle \( \theta \) in standard position, \( \tan(\theta) = \frac{y}{x} \), where \( y \) is the opposite side (y-coordinate) and \( x \) is the adjacent side (x-coordinate).

Step2: Substitute values

From the diagram, \( y = 2.31 \, \text{cm} \), \( x = 3.8 \, \text{cm} \).
\( \tan(\theta) = \frac{2.31}{3.8} \approx 0.6079 \)

Part (a) Answer:

The radius \( r \approx \boldsymbol{4.45 \, \text{cm}} \) (or more precisely, \( \sqrt{19.7761} = 4.447 \, \text{cm} \approx 4.45 \, \text{cm} \))

Part (b) Answer:

\( \cos(\theta) \approx \boldsymbol{0.855} \) (or more precisely, \( \frac{3.8}{4.447} \approx 0.8545 \))

Part (c) Answer:

\( \tan(\theta) \approx \boldsymbol{0.608} \) (or more precisely, \( \frac{2.31}{3.8} \approx 0.6079 \))