Question

Sketch an angle θ in standard position such that θ has the least possible positive measure, and the point (-2√3,-2) is on the terminal side of θ. Then find the exact values of the six trigonometric functions for θ. Rationalize denominators when aplicable.​

Answer 1

Answer:

See below  

Step-by-step explanation:

We can use the unit circle to figure out the trigonometric functions of θ.

Both the x- and y- coordinates are in the third quadrant, so θ is an obtuse angle (210°), as seen in the figure below.

sinθ = -2/4         = -½

cosθ = (-2√3)/4  = -(√3)/2

tanθ = -2/(-2√3) = 1/√3         = (√3)/3

cscθ = 1/sin θ     = 4/(-2)        = -2

secθ = 1/cosθ     = 4/(-2√3)  = -2/√3 = -(2√3)/3

cotθ = 1/tan θ     = -2√3/(-2) = √3

Answer 2

The exact values of the six trigonometric functions are: $\sin \theta = -\frac{1}{2}$, $\cos \theta = -\frac{\sqrt{3}}{4}$, $\tan \theta = \frac{\sqrt{3}}{3}$, $\cot \theta = \sqrt{3}$, $\sec \theta = -\frac{2\sqrt{3}}{3}$, $\csc \theta = -2$.

An angle in standard position is an angle measured with respect to the +x semiaxis. In this question we must derive the six trigonometric functions from the distances between a given point and the origin. The trigonometric functions are described below:

Sine

$\sin \theta = \frac{y}{\sqrt{x^{2}+y^{2}}}$ (1)

Cosine

$\cos \theta = \frac{x}{\sqrt{x^{2}+y^{2}}}$ (2)

Tangent

$\tan\theta = \frac{y}{x}$ (3)

Cotangent

$\cot \theta = \frac{x}{y}$ (4)

Secant

$\sec \theta = \frac{\sqrt{x^{2}+y^{2}}}{x}$ (5)

Cosecant

$\csc\theta = \frac{\sqrt{x^{2}+y^{2}}}{y}$ (6)

Where $\theta$ is the terminal angle in sexagesimal degrees.

If we know that $x = -2\sqrt{3}$ and $y = -2$, then the exact values of the six trigonometric functions are:

$\sin \theta = \frac{-2}{4}$, $\cos \theta = -\frac{2\sqrt{3}}{4}$, $\tan \theta = \frac{2}{2\sqrt{3}}$, $\cot \theta = \frac{2\sqrt{3}}{2}$, $\sec \theta = -\frac{4}{2\sqrt{3}}$, $\csc \theta = -\frac{4}{2}$

$\sin \theta = -\frac{1}{2}$, $\cos \theta = -\frac{\sqrt{3}}{4}$, $\tan \theta = \frac{\sqrt{3}}{3}$, $\cot \theta = \sqrt{3}$, $\sec \theta = -\frac{2\sqrt{3}}{3}$, $\csc \theta = -2$

The exact values of the six trigonometric functions are: $\sin \theta = -\frac{1}{2}$, $\cos \theta = -\frac{\sqrt{3}}{4}$, $\tan \theta = \frac{\sqrt{3}}{3}$, $\cot \theta = \sqrt{3}$, $\sec \theta = -\frac{2\sqrt{3}}{3}$, $\csc \theta = -2$.

We kindly invite to see this question on trigonometric functions: https://brainly.com/question/6904750