# The terminal side of #theta# lies on the line #2x-y=0# in quadrant III, how do you find the values of the six trigonometric functions by finding a point on the line?

Any relation of the form

So

Furthermore, we can see that

so we have the situation in the image below for a point on the line in Q III:

By signing up, you agree to our Terms of Service and Privacy Policy

The answer is:

Because

Using the distance formula you arrive at the square root of 5. This answer is verified via a Larson Precalculus book in the answer section.

By signing up, you agree to our Terms of Service and Privacy Policy

To find the values of the six trigonometric functions for an angle ( \theta ) whose terminal side lies on the line ( 2x - y = 0 ) in quadrant III, we need a point on the line that also lies in quadrant III, where both ( x ) and ( y ) are negative.

The equation ( 2x - y = 0 ) can be rewritten as ( y = 2x ). We need a point ((x, y)) on this line in the third quadrant. A simple choice would be ( x = -1 ), which gives ( y = -2 ) (since ( y = 2(-1) = -2 )).

So, we have the point ((-1, -2)) on the line in quadrant III. This point means that if you draw a right triangle from the origin to this point, the lengths of the sides relative to the origin would be:

- ( x = -1 ) (the horizontal distance, or the adjacent side of the angle ( \theta )),
- ( y = -2 ) (the vertical distance, or the opposite side of the angle ( \theta )),
- The hypotenuse ( r ) can be found using the Pythagorean theorem: ( r = \sqrt{x^2 + y^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} ).

The six trigonometric functions of ( \theta ) are defined as follows:

- Sine (( \sin(\theta) )) is opposite/hypotenuse: ( \sin(\theta) = \frac{-2}{\sqrt{5}} ).
- Cosine (( \cos(\theta) )) is adjacent/hypotenuse: ( \cos(\theta) = \frac{-1}{\sqrt{5}} ).
- Tangent (( \tan(\theta) )) is opposite/adjacent: ( \tan(\theta) = \frac{-2}{-1} = 2 ).
- Cosecant (( \csc(\theta) )) is the reciprocal of sine: ( \csc(\theta) = \frac{\sqrt{5}}{-2} ).
- Secant (( \sec(\theta) )) is the reciprocal of cosine: ( \sec(\theta) = \frac{\sqrt{5}}{-1} ).
- Cotangent (( \cot(\theta) )) is the reciprocal of tangent: ( \cot(\theta) = \frac{1}{2} ).

These values are based on the point ((-1, -2)) and respect the signs of trigonometric functions in quadrant III, where both sine and cosine are negative, and consequently, their reciprocals are also negative.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the side lengths for a right triangle where only the angles of 90, 18, and 72 degrees are given?
- How do you find the value of #sin^2(225^circ)#?
- From the top of a hill, the angles of depression of two consecutive kilometer stones due to east are found to be 30° and 45°. Find the height of the hill (or) what is the height of the hill?
- Using the graphs to find the value of x, is sin x cos x > 1 a true statement?
- What is the length of the hypotenuse of a right triangle if the two other sides are of lengths 9 and 1?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7