Evaluate all six trigonometric functions at t, where the given point lies on the terminal side of an angle of t radians in standard position. (40/41, -9/41)
To evaluate the trigonometric functions at the point (40/41, -9/41), we need to first find the values of sine, cosine, tangent, cosecant, secant, and cotangent. Here's how we can do that:
1. Start by finding the hypotenuse of the right triangle formed by the given point. The hypotenuse is the distance from the origin (0, 0) to the given point.
To find the hypotenuse:
Hypotenuse = sqrt((40/41)^2 + (-9/41)^2)
Hypotenuse = sqrt(1600/1681 + 81/1681)
Hypotenuse = sqrt(1681/1681)
Hypotenuse = 1
2. Next, we need to find the values of the trigonometric functions using the ratios of the sides of the right triangle. Since we have the coordinates of the point, we can determine the lengths of the adjacent side and opposite side of the triangle.
Adjacent side = 40/41
Opposite side = -9/41
Now, we can evaluate the trigonometric functions:
- Sine (sin): Sine is the ratio of the opposite side to the hypotenuse.
sin(t) = Opposite side / Hypotenuse = (-9/41) / 1 = -9/41
- Cosine (cos): Cosine is the ratio of the adjacent side to the hypotenuse.
cos(t) = Adjacent side / Hypotenuse = (40/41) / 1 = 40/41
- Tangent (tan): Tangent is the ratio of the opposite side to the adjacent side.
tan(t) = Opposite side / Adjacent side = (-9/41) / (40/41) = -9/40
- Cosecant (csc): Cosecant is the reciprocal of sine.
csc(t) = 1 / sin(t) = 1 / (-9/41) = -41/9
- Secant (sec): Secant is the reciprocal of cosine.
sec(t) = 1 / cos(t) = 1 / (40/41) = 41/40
- Cotangent (cot): Cotangent is the reciprocal of tangent.
cot(t) = 1 / tan(t) = 1 / (-9/40) = -40/9
So, the values of the trigonometric functions at t (where the given point lies on the terminal side of an angle of t radians) are:
sin(t) = -9/41
cos(t) = 40/41
tan(t) = -9/40
csc(t) = -41/9
sec(t) = 41/40
cot(t) = -40/9
To evaluate the trigonometric functions at a given point (x, y) on the terminal side of an angle in standard position, we need to find the values of the sine, cosine, tangent, cosecant, secant, and cotangent functions.
Let's start by finding the values of the sine and cosine functions:
1. Sine (sin(t)):
Sine is the y-coordinate of the given point divided by the hypotenuse. In this case, the y-coordinate is -9/41. The hypotenuse can be found using Pythagorean theorem, which is the square root of the sum of the squares of the x and y coordinates: √((40/41)^2 + (-9/41)^2). Simplifying this gives a hypotenuse of 1.
So, sin(t) = -9/41 / 1 = -9/41.
2. Cosine (cos(t)):
Cosine is the x-coordinate of the given point divided by the hypotenuse. In this case, the x-coordinate is 40/41.
So, cos(t) = 40/41 / 1 = 40/41.
Next, we can find the values of the other four trigonometric functions using the definitions and reciprocal properties:
3. Tangent (tan(t)):
Tangent is the sine divided by the cosine. So, tan(t) = sin(t) / cos(t) = (-9/41) / (40/41). This simplifies to -9/40.
4. Cosecant (csc(t)):
Cosecant is the reciprocal of sine. So, csc(t) = 1 / sin(t) = 1 / (-9/41). This simplifies to -41/9.
5. Secant (sec(t)):
Secant is the reciprocal of cosine. So, sec(t) = 1 / cos(t) = 1 / (40/41). This simplifies to 41/40.
6. Cotangent (cot(t)):
Cotangent is the reciprocal of tangent. So, cot(t) = 1 / tan(t) = 1 / (-9/40). This simplifies to -40/9.
To summarize:
sin(t) = -9/41
cos(t) = 40/41
tan(t) = -9/40
csc(t) = -41/9
sec(t) = 41/40
cot(t) = -40/9