the terminal side of an angle theta in standard position coincides with the line y=5x and lies in quadrant 3. find the six trigonometric functions of theta.
Y = 5x.
m = 5 = Tan A, A = 258.7o.
Sin A = -0.98058.
Cos A = -0.19612.
Tan A = 5.00000.
Csc A = -1.01980.
SEC A = -5.09902.
COT A = 0.20000.
Simplify the trigonometric expression.
sin^2 theta / 1-cos theta
We can use the identity 1 - cos^2(theta) = sin^2(theta) to simplify the given expression:
sin^2(theta) / (1 - cos(theta))
= sin^2(theta) / [(1 - cos(theta))(1 + cos(theta))]
= sin^2(theta) / (1 - cos^2(theta))
= sin^2(theta) / sin^2(theta)
= 1
Therefore, sin^2(theta) / (1 - cos(theta)) simplifies to 1.
To find the six trigonometric functions of an angle, we first need to determine the coordinates of the point where the terminal side of the angle intersects with the unit circle in standard position.
Let's start by determining the coordinates of the point of intersection between the line y = 5x and the unit circle. Since the line passes through the origin (0, 0), we need to find the point on the line that is one unit away from the origin. We can do this by finding the value of x when y = 5x is equal to 1.
5x = 1
x = 1/5
So, the coordinates of the point of intersection are (1/5, 1). Since the terminal side of the angle lies in quadrant 3, the value of x will be negative. Therefore, the coordinates of the point on the terminal side are (-1/5, 1).
Now, we can find the trigonometric functions of the angle theta.
1. Sine (sin): The sine of an angle is the y-coordinate of the point on the unit circle. Therefore, sin(theta) = y-coordinate = 1.
2. Cosine (cos): The cosine of an angle is the x-coordinate of the point on the unit circle. Therefore, cos(theta) = x-coordinate = -1/5.
3. Tangent (tan): The tangent of an angle is the ratio of sine to cosine. Therefore, tan(theta) = sin(theta) / cos(theta) = 1 / (-1/5) = -5.
4. Cosecant (csc): The cosecant of an angle is the reciprocal of the sine. Therefore, csc(theta) = 1 / sin(theta) = 1 / 1 = 1.
5. Secant (sec): The secant of an angle is the reciprocal of the cosine. Therefore, sec(theta) = 1 / cos(theta) = 1 / (-1/5) = -5.
6. Cotangent (cot): The cotangent of an angle is the reciprocal of the tangent. Therefore, cot(theta) = 1 / tan(theta) = 1 / (-5) = -1/5.
So, the six trigonometric functions of the angle theta are:
sin(theta) = 1
cos(theta) = -1/5
tan(theta) = -5
csc(theta) = 1
sec(theta) = -5
cot(theta) = -1/5