Find the values of the six trigonometric functions for ∠A in standard position determined by the point (63, 280). Enter your answers as fractions.
The point is in the 1st quadrant, so all our ratios will be positive
r^2 = 63^ + 280^2 = 82369
r =√82369 = 287
sinA = y/r = 280/287 = 40/41 , then cscA = 41/40
cosA = x/r = 63/287 = 9/41 , then secA = 41/9
tanA = y/x = 280/63 = 40/9 , then cotA = 9/40
To find the values of the six trigonometric functions for angle A, we first need to determine the ratios of the sides of a right triangle formed by the given point (63, 280). Let's denote the opposite side as y and the adjacent side as x.
Using the Pythagorean theorem, we can find the hypotenuse (r) of the triangle:
r = √(x^2 + y^2) = √(63^2 + 280^2)
Now, let's find the values of the trigonometric functions:
1. Sine (sin):
sin(A) = y/r = 280/√(63^2 + 280^2)
2. Cosine (cos):
cos(A) = x/r = 63/√(63^2 + 280^2)
3. Tangent (tan):
tan(A) = y/x = 280/63
4. Cosecant (csc):
csc(A) = 1/sin(A)
5. Secant (sec):
sec(A) = 1/cos(A)
6. Cotangent (cot):
cot(A) = 1/tan(A)
Please note that there might be an approximation in the values since √(63^2 + 280^2) involves a square root.
To find the values of the six trigonometric functions for ∠A, we can use the coordinates of the point (63, 280) to determine the lengths of the sides of a right triangle.
Firstly, let's find the length of the hypotenuse (r) using the Pythagorean theorem. The hypotenuse is the distance from the origin to the point (63, 280).
r = √((x^2) + (y^2))
= √((63^2) + (280^2))
= √(3969 + 78400)
= √82369
= 287 (approximately)
Next, we need to determine the values of the two sides of the triangle relative to the angle ∠A.
The x-coordinate of the point (63, 280) represents the adjacent side (x) of the triangle.
x = 63
The y-coordinate of the point (63, 280) represents the opposite side (y) of the triangle.
y = 280
Now, we can calculate the values of the six trigonometric functions for ∠A.
1. Sin (∠A) = y / r
= 280 / 287
= 280/287
2. Cos (∠A) = x / r
= 63 / 287
= 63/287
3. Tan (∠A) = Sin (∠A) / Cos(∠A)
= (280/287) / (63/287)
= 280/63
= 40/9
4. Cosec (∠A) = 1 / Sin (∠A)
= 1 / (280/287)
= 240/280
= 6/7
5. Sec (∠A) = 1 / Cos (∠A)
= 1 / (63/287)
= 287/63
= (9 * 287) / (9 * 63)
= 287/63
6. Cot (∠A) = 1 / Tan (∠A)
= 1 / (40/9)
= 9/40
Therefore, the values of the six trigonometric functions for ∠A are:
Sin (∠A) = 280/287
Cos (∠A) = 63/287
Tan (∠A) = 40/9
Cosec (∠A) = 6/7
Sec (∠A) = 287/63
Cot (∠A) = 9/40