P is a point on the terminal side of θ in standard position. Find the exact value of ALL SIX trigonometric functions. P (5, −12)
r = 13. Now consider that
sinθ = y/r
cosθ = x/r
tanθ = y/x
and just crank them out
what about the other three? also how do you find what r is
and thank you so much you have really been of great help today
my teacher really doesnt teach well and i dont understand trig at all so thank you so much
r^2 = x^2 + y^2
the other three are the reciprocals of these.
Looks like you need to review the topic.
To find the exact value of all six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for the point P (5, -12) in standard position, we can use the Pythagorean theorem to find the length of the hypotenuse, and then apply the definitions of the trigonometric functions.
First, let's label the sides of the right triangle formed by the coordinates of P (5, -12) with respect to the origin (0,0):
The horizontal side adjacent to the angle θ is the x-coordinate, which is 5.
The vertical side opposite to the angle θ is the y-coordinate, which is -12.
The hypotenuse, denoted as r, can be found using the Pythagorean theorem:
r² = (adjacent side)² + (opposite side)²
r² = 5² + (-12)²
r² = 25 + 144
r² = 169
r = √169
r = 13
Now that we have the hypotenuse length (r), we can calculate the trigonometric functions of θ:
1. Sine (sin): sin(θ) = opposite/hypotenuse
sin(θ) = -12/13
2. Cosine (cos): cos(θ) = adjacent/hypotenuse
cos(θ) = 5/13
3. Tangent (tan): tan(θ) = opposite/adjacent
tan(θ) = -12/5
4. Cosecant (csc): csc(θ) = 1/sin(θ)
csc(θ) = 1/(-12/13)
csc(θ) = -13/12
5. Secant (sec): sec(θ) = 1/cos(θ)
sec(θ) = 1/(5/13)
sec(θ) = 13/5
6. Cotangent (cot): cot(θ) = 1/tan(θ)
cot(θ) = 1/(-12/5)
cot(θ) = -5/12
Therefore, the exact values of all six trigonometric functions for the point P (5, -12) are:
sin(θ) = -12/13
cos(θ) = 5/13
tan(θ) = -12/5
csc(θ) = -13/12
sec(θ) = 13/5
cot(θ) = -5/12