Question

When Θ = 5 pi over 6, what are the reference angle and the sign values for sine, cosine, and tangent?

Θ' = negative pi over 6; sine and cosine are positive, tangent is negative.
Θ' = 5 pi over 6; sine and tangent are positive, cosine is negative
Θ' = pi over 6; sine is positive, cosine and tangent are negative
Θ' = negative 5 pi over 6; sine is positive, cosine and tangent are negative

Please someone help me. Thank you.

Answer 1

Note that (5π)/6 radians = 150°. Therefore the given angle is in quadrant 2.

Refer to the figure shown below.

Reference angles are measured relative to the horizontal axis.
Therefore the reference angle in each quadrant is π/6 radians or 30°.
Denote the reference angle as θ'.
Then, in quadrant 1,
cos θ' = √3/2,  sin θ' = 1/2,  tan θ' = √3.

Because we are in quadrant 2,
  sin θ' = π/6;
  sin(5π/6) is positive, but cos (5π/6) and tan (5π/6) are negative.

 Answer:
5π/6 is in quadrant 2.
The reference angle, θ' = π/6.
sin(5π/6) is positive, cosine and tangent are negative.

Answer 2

You can use the fact that given angle lies in second quadrant where only sin and cosine are positive and rest of the trigonometric functions output negative values.

Option C: Θ' = pi over 6; sine is positive, cosine and tangent are negative is correct.

How to find if the angle given lies in which quadrant?

• If angle lies between 0 to $\dfrac{\pi}{2}$, then it is int first quadrant.
• If angle lies between $\dfrac{\pi}{2}$ to a $\pi$, then it is in second quadrant.
• When the angle lies between  $\pi$ and $\dfrac{3\pi}{2}$, then that angle lies in 3rd quadrant.
• And when it lies from $\dfrac{3\pi}{2}$  and 0 degrees, then the angle is in fourth quadrant.

Which trigonometric functions are positive in which quadrant?

• In first quadrant, all six trigonometric functions are positive.
• In second quadrant, only sin and cosec are positive.
• In the third quadrant, only tangent and cotangent are positive.
• In fourth, only cos and sec are positive.

(this all positive negative refers to the fact that if you use given angle  as input to these functions, then what sign will these functions will evaluate based on in which quadrant does the given angle lies.)

Since we have:

$\dfrac{\pi}{2} < \dfrac{5\pi}{6} < \pi$, thus the given angles lies in second quadrant.

The reference angle will be supplement(180 degrees - that angle) of the given angle since it lies in second quadrant.

$\text{Reference angle} = \pi - \dfrac{5\pi}{6} = \dfrac{\pi}{6}$ (as 180 degrees is written pi radians)

Since in second quadrant only sine and cosec are positive, and rest are negative.

Thus Option C: Θ' = pi over 6; sine is positive, cosine and tangent are negative is correct.

Learn more about quadrants and signs of trigonometric functions here:

https://brainly.com/question/12380695