Can someone help me figure these out? I think if someone can explain how to do one of them then I'll be able to do the other. please and thank you.
Suppose θ is an angle in standard position whose terminal side is in the given quadrant. For each function, find the exact values of the remaining five trigonometric functions of θ.
sin θ= 4/5, Quadrant 2
tan θ= -12/5, Quadrant 4
Certainly! Let's start with finding the values of the remaining trigonometric functions for θ when sin θ = 4/5 and θ is in Quadrant 2.
In Quadrant 2, the x-coordinate is negative and the y-coordinate is positive. To find the values of the remaining trigonometric functions, we can use the right triangle formed by the angle θ in standard position.
Here's how you can find the remaining trigonometric functions:
1. Start by finding the cosine, which is adjacent side divided by hypotenuse.
cos θ = adjacent side / hypotenuse
Since θ is in Quadrant 2, the adjacent side is negative and can be determined using the Pythagorean identity.
We can use the Pythagorean identity: sin^2θ + cos^2θ = 1.
sin θ = 4/5.
Let's find cos θ:
sin^2θ + cos^2θ = 1
(4/5)^2 + cos^2θ = 1
16/25 + cos^2θ = 1
cos^2θ = 1 - 16/25
cos^2θ = 9/25
Taking the square root of both sides:
cos θ = ±√(9/25)
cos θ = ±3/5
Since θ is in Quadrant 2, the cosine is negative:
cos θ = -3/5
2. Next, find the tangent, which is opposite side divided by adjacent side.
tan θ = opposite side / adjacent side
Given that tan θ = -12/5, we can find the opposite side using the ratio:
tan θ = opposite side / adjacent side
-12/5 = opposite side / (-3/5)
Multiplying both sides by (-3/5):
-12/5 * (-3/5) = opposite side
opposite side = 36/25
Therefore, the opposite side is 36/25.
3. With the opposite side and adjacent side known, we can find the remaining trigonometric functions:
sec θ = hypotenuse / adjacent side
sec θ = 1 / cos θ
Since cos θ = -3/5, we can substitute it in:
sec θ = 1 / (-3/5)
sec θ = -5/3
csc θ = hypotenuse / opposite side
csc θ = 1 / sin θ
Since sin θ = 4/5, we can substitute it in:
csc θ = 1 / (4/5)
csc θ = 5/4
cot θ = adjacent side / opposite side
cot θ = -3/5 / (36/25)
Multiplying by the reciprocal:
cot θ = -3/5 * (25/36)
cot θ = -15/12
Simplifying, we get cot θ = -5/4.
Therefore, the exact values of the remaining five trigonometric functions in Quadrant 2, when sin θ = 4/5, are:
cos θ = -3/5
sec θ = -5/3
csc θ = 5/4
tan θ = -12/5
cot θ = -5/4
Now let's move on to the second question, finding the values of the remaining trigonometric functions for θ when tan θ = -12/5 and θ is in Quadrant 4.