The given point lies on the terminal side of an angle ϴ in standard position. Find the value of the six trigonometric functions of ϴ.(2, -4)
Draw the point on the cartesian plane, then use the pythagorean theorem to find the longest side of the triangle made with the x-axis, then use your primary trig ratios first.
for instance sintheta = opp/hyp
To find the value of the six trigonometric functions of ϴ given the point (2, -4) lies on the terminal side of the angle in standard position, we need to determine the values of the coordinates of the point (x, y) and use them in the trigonometric functions.
First, let's find the values of x and y. In this case, x = 2 and y = -4.
The six trigonometric functions are defined as follows:
1. Sine (sinϴ): Sine is the ratio of the y-coordinate to the radius of the unit circle. We can calculate it by dividing the y-coordinate by the radius. The radius of the unit circle is always 1, so sinϴ = y / 1 = y.
Therefore, sinϴ = -4.
2. Cosine (cosϴ): Cosine is the ratio of the x-coordinate to the radius of the unit circle. We can calculate it by dividing the x-coordinate by the radius. Again, the radius of the unit circle is always 1, so cosϴ = x / 1 = x.
Therefore, cosϴ = 2.
3. Tangent (tanϴ): Tangent is the ratio of the sine to the cosine. Therefore, tanϴ = sinϴ / cosϴ.
Substituting the values we found, tanϴ = sinϴ / cosϴ = (-4) / 2 = -2.
4. Cosecant (cscϴ): Cosecant is the reciprocal of sine. Therefore, cscϴ = 1 / sinϴ.
Substituting the value we found, cscϴ = 1 / sinϴ = 1 / (-4) = -1/4.
5. Secant (secϴ): Secant is the reciprocal of cosine. Therefore, secϴ = 1 / cosϴ.
Substituting the value we found, secϴ = 1 / cosϴ = 1 / 2 = 1/2.
6. Cotangent (cotϴ): Cotangent is the reciprocal of tangent. Therefore, cotϴ = 1 / tanϴ.
Substituting the value we found, cotϴ = 1 / tanϴ = 1 / (-2) = -1/2.
To summarize:
sinϴ = -4
cosϴ = 2
tanϴ = -2
cscϴ = -1/4
secϴ = 1/2
cotϴ = -1/2