Find sin t ,cos t and tan t when the terminal side of an angle of t radians passes through (2 , 7)
slope= 7/2
so draw a triangle wiht sides 2,7,sqrt53
sinT=7/sqrt53
cosT=2/sqrt53
you do tan.
To find sin(t), cos(t), and tan(t), we can use the given information that the terminal side of angle t passes through the point (2, 7) in the Cartesian coordinate plane.
We can find the values of sin(t) and cos(t) using the coordinates (x, y) of the point where the terminal side passes through. Here, x = 2 and y = 7.
Step 1: Find the value of the hypotenuse (r) using the Pythagorean theorem:
r = sqrt(x^2 + y^2)
= sqrt(2^2 + 7^2)
= sqrt(4 + 49)
= sqrt(53)
Step 2: Find the value of sin(t) using the equation:
sin(t) = y / r
= 7 / sqrt(53)
To find an approximate value, we can divide 7 by the square root of 53:
sin(t) ≈ 0.961
Step 3: Find the value of cos(t) using the equation:
cos(t) = x / r
= 2 / sqrt(53)
To find an approximate value, we can divide 2 by the square root of 53:
cos(t) ≈ 0.276
Step 4: Find the value of tan(t) using the equation:
tan(t) = y / x
= 7 / 2
To find an approximate value, we can divide 7 by 2:
tan(t) ≈ 3.5
Therefore, sin(t) ≈ 0.961, cos(t) ≈ 0.276, and tan(t) ≈ 3.5.
To find the values of sine, cosine, and tangent of an angle that passes through a specific point (x, y), you can use the following steps:
1. Find the hypotenuse of the right triangle formed by the given point (x, y) and the origin (0, 0). The hypotenuse is the distance between the origin and the point (x, y).
In this case, the distance between the origin (0, 0) and (2, 7) can be calculated using the Pythagorean theorem:
hypotenuse = sqrt(2^2 + 7^2) = sqrt(4 + 49) = sqrt(53)
2. Determine the values of sine, cosine, and tangent using the coordinates of the point (x, y) and the hypotenuse.
- The sine of the angle (sin t) is equal to the opposite side divided by the hypotenuse. In this case, the opposite side is 7, and the hypotenuse is sqrt(53).
sin t = 7 / sqrt(53)
- The cosine of the angle (cos t) is equal to the adjacent side divided by the hypotenuse. In this case, the adjacent side is 2, and the hypotenuse is sqrt(53).
cos t = 2 / sqrt(53)
- The tangent of the angle (tan t) is equal to the opposite side divided by the adjacent side. In this case, the opposite side is 7, and the adjacent side is 2.
tan t = 7 / 2
Therefore, sin t = 7 / sqrt(53), cos t = 2 / sqrt(53), and tan t = 7 / 2.