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
To find the exact values of the remaining trigonometric functions of θ, we can use the given information about the quadrant and the trigonometric identities. Let's start with finding the values for the first problem.
sin θ = 4/5, Quadrant 2:
In Quadrant 2, the sine function is positive, and other trigonometric functions will have specific signs based on quadrant locations. To find the remaining trigonometric functions, we can use the Pythagorean identity:
sin^2 θ + cos^2 θ = 1
Substituting the given value of sin θ:
(4/5)^2 + cos^2 θ = 1
16/25 + cos^2 θ = 1
cos^2 θ = 1 - 16/25
cos^2 θ = 9/25
Taking the square root of both sides:
cos θ = ± √(9/25)
cos θ = ± (3/5)
Since θ is in Quadrant 2, where the cosine function is negative, we take the negative value:
cos θ = -3/5
Now, let's find the remaining trigonometric functions of θ:
1. tan θ = sin θ / cos θ
tan θ = (4/5) / (-3/5)
tan θ = -4/3
2. csc θ = 1 / sin θ
csc θ = 1 / (4/5)
csc θ = 5/4
3. sec θ = 1 / cos θ
sec θ = 1 / (-3/5)
sec θ = -5/3
4. cot θ = 1 / tan θ
cot θ = 1 / (-4/3)
cot θ = -3/4
Therefore, for the given values of sin θ = 4/5 in Quadrant 2, the remaining trigonometric functions of θ are:
cos θ = -3/5
tan θ = -4/3
csc θ = 5/4
sec θ = -5/3
cot θ = -3/4
Now, let's move on to the second problem.
tan θ = -12/5, Quadrant 4:
In Quadrant 4, the tangent function is positive, and other trigonometric functions will have specific signs based on quadrant locations. Let's use the Pythagorean identity again:
tan^2 θ + 1 = sec^2 θ
Substituting the given value of tan θ:
(-12/5)^2 + 1 = sec^2 θ
144/25 + 1 = sec^2 θ
169/25 = sec^2 θ
Taking the square root of both sides:
sec θ = ± √(169/25)
sec θ = ± (13/5)
Since θ is in Quadrant 4, where the secant function is positive, we take the positive value:
sec θ = 13/5
Now, let's find the remaining trigonometric functions:
1. sin θ = tan θ / sec θ
sin θ = (-12/5) / (13/5)
sin θ = -12/13
2. cos θ = 1 / sec θ
cos θ = 1 / (13/5)
cos θ = 5/13
3. csc θ = 1 / sin θ
csc θ = 1 / (-12/13)
csc θ = -13/12
4. cot θ = 1 / tan θ
cot θ = 1 / (-12/5)
cot θ = -5/12
Therefore, for the given values of tan θ = -12/5 in Quadrant 4, the remaining trigonometric functions of θ are:
sin θ = -12/13
cos θ = 5/13
csc θ = -13/12
sec θ = 13/5
cot θ = -5/12
I hope this explanation helps you understand how to find the values of trigonometric functions in different quadrants. If you have any further questions or need additional assistance, feel free to ask!
Of course, I can help you with these trigonometric functions!
To find the exact values of the remaining five trigonometric functions of θ, we will need to remember the identities:
- sine (sin) is positive in Quadrant 2
- cosine (cos) is negative in Quadrant 2
- tangent (tan) is negative in Quadrant 2
- sine (sin) is negative in Quadrant 4
- cosine (cos) is positive in Quadrant 4
- tangent (tan) is negative in Quadrant 4
Let's proceed with the first case: sin θ = 4/5 in Quadrant 2.
1. We know that sin θ = opposite/hypotenuse, so we have the opposite side as 4 and the hypotenuse as 5.
2. Using the Pythagorean theorem, we can find the adjacent side:
a^2 + b^2 = c^2
a^2 + 4^2 = 5^2
a^2 = 25 - 16
a^2 = 9
a = √9
a = 3
Therefore, in Quadrant 2, the exact values of the trigonometric functions for sin θ = 4/5 are:
sin θ = 4/5
cos θ = -√(1 - sin^2 θ) = -√(1 - (4/5)^2) = -√(1 - 16/25) = -√(9/25) = -3/5
tan θ = sin θ/cos θ = (4/5) / (-3/5) = -4/3
csc θ = 1/sin θ = 1 / (4/5) = 5/4
sec θ = 1/cos θ = 1 / (-3/5) = -5/3
cot θ = 1/tan θ = 1 / (-4/3) = -3/4
Now let's move on to the second case: tan θ = -12/5 in Quadrant 4.
1. We know that tan θ = opposite/adjacent, so we have the opposite side as -12 and the adjacent side as 5.
2. Using the Pythagorean theorem, we can find the hypotenuse:
a^2 + (-12)^2 = 5^2
a^2 + 144 = 25
a^2 = 25 - 144
a^2 = -119
Since a^2 is negative, it is not a valid length, which means there is no triangle in Quadrant 4 that satisfies tan θ = -12/5. Therefore, the remaining trigonometric functions cannot be determined.
To summarize, for tan θ = -12/5 in Quadrant 4, the remaining five trigonometric functions cannot be determined as there is no such triangle in that quadrant.
I hope this explanation helps you understand how to find the exact values of trigonometric functions in different quadrants. If you have any further questions, feel free to ask!