Question

Let θ be an angle in the first quadrant, and suppose sin(θ)=a. Evaluate the following expressions in terms of a. For example, sin(θ+180∘)=−a. Your answers will be expressions that involve the letter a. Sketch a picture of the angles to help.

sin(θ+360∘)=

Answer 1

To evaluate sin(θ+360°) when sin(θ)=a, we use the periodic property of the sine function. Since the sine function repeats every 360°, sin(θ+360°) is equal to sin(θ), which is a.

Step-by-step explanation:

Let θ be an angle in the first quadrant, and suppose sin(θ)=a. To evaluate sin(θ+360°), we can use the periodic property of the sine function, which states that sine is periodic with a period of 360°. This means that sin(θ+360°) = sin(θ), because adding 360° to an angle results in a full circle, bringing us back to the original angle θ within the unit circle.

To evaluate the expression sin(θ+360∘) in terms of a, we can use the angle addition formula for sine:

sin(θ+360∘) = sin(θ)cos(360∘) + cos(θ)sin(360∘)

Since cos(360∘) = 1 and sin(360∘) = 0, the expression simplifies to:

sin(θ+360∘) = sin(θ)

Therefore, sin(θ+360∘) is equal to a. This means that no matter what value θ takes in the first quadrant, adding 360∘ to it does not change its sine value.

Therefore, sin(θ+360°) = a, same as the original sin(θ).