If sinθ=7/13 and cosθ=12/13 find tan θ and cot θ
Use Pythagorean Identities to find sin θ and tan θ if cos θ =-24/25 if the terminal side of θ lies in the third quadrant.
hellooo help me please
recall that for the point (x,y)
sinθ = y/r
cosθ = x/r
tanθ = y/x
So, for your two problems,
If sinθ=7/13 and cosθ=12/13 find tan θ and cot θ
Unfortunately, no angle θ has sinθ=7/13 and cosθ=12/13 since 7^2+12^2 ≠ 13^2
However, 5^2+12^2 = 13^2, so if
sinθ=5/13 and cosθ=12/13
then tanθ = sinθ/cosθ = 5/12
cos θ =-24/25 in QIII, so
x = -24
y = -7
r = 25
So,
sinθ = y/r = -7/25
tanθ = y/x = 7/24
To find tan θ and cot θ, we can use the given values of sin θ and cos θ.
Given sin θ = 7/13 and cos θ = 12/13:
1. Remember that tan θ is equal to sin θ divided by cos θ:
tan θ = sin θ / cos θ
2. Substitute the values of sin θ = 7/13 and cos θ = 12/13 into the formula:
tan θ = (7/13) / (12/13)
3. Simplify the expression:
tan θ = 7/13 * 13/12
= 7/12
Therefore, tan θ = 7/12.
4. To find cot θ, we can use the reciprocal of tan θ:
cot θ = 1 / tan θ
5. Substitute the value of tan θ = 7/12 into the formula:
cot θ = 1 / (7/12)
6. Simplify the expression:
cot θ = 12/7
Therefore, cot θ = 12/7.
Now, let's use the Pythagorean Identity to find sin θ and tan θ when cos θ = -24/25 and θ lies in the third quadrant.
1. Recall the Pythagorean Identity:
sin² θ + cos² θ = 1
2. Substitute the given value of cos θ = -24/25 into the identity:
sin² θ + (-24/25)² = 1
sin² θ + 576/625 = 1
sin² θ = 1 - 576/625
= (625 - 576) / 625
= 49 / 625
3. Take the square root of both sides to find sin θ:
sin θ = √(49/625)
= √49 / √625
= 7/25
Therefore, sin θ = 7/25.
4. To find tan θ, we can use the definition of tan θ:
tan θ = sin θ / cos θ
5. Substitute the values of sin θ = 7/25 and cos θ = -24/25 into the formula:
tan θ = (7/25) / (-24/25)
6. Simplify the expression:
tan θ = (7/25) * (-25/24)
= -7/24
Therefore, tan θ = -7/24.
To find tan θ and cot θ, we need to use the following formulas:
1. tan θ = sin θ / cos θ
2. cot θ = cos θ / sin θ
Given that sin θ = 7/13 and cos θ = 12/13, let's substitute these values into the formulas to find the values of tan θ and cot θ.
1. tan θ = sin θ / cos θ
tan θ = (7/13) / (12/13)
tan θ = 7/13 * 13/12
tan θ = 7/12, which is the value of tan θ.
2. cot θ = cos θ / sin θ
cot θ = (12/13) / (7/13)
cot θ = 12/13 * 13/7
cot θ = 12/7, which is the value of cot θ.
Therefore, tan θ = 7/12 and cot θ = 12/7.
Now, let's move on to the second question.
To find sin θ and tan θ when cos θ = -24/25 and the terminal side of θ lies in the third quadrant, we can use the Pythagorean Identity: sin² θ + cos² θ = 1.
Given cos θ = -24/25, we can find sin θ using the Pythagorean Identity.
1. sin² θ + cos² θ = 1
sin² θ + (-24/25)² = 1
sin² θ + 576/625 = 1
sin² θ = 1 - 576/625
sin² θ = 625/625 - 576/625
sin² θ = 49/625
sin θ = √(49/625)
Since θ lies in the third quadrant (cos θ is negative), sin θ will also be negative. Therefore, sin θ = -7/25.
To find tan θ, we can use the formula:
2. tan θ = sin θ / cos θ
tan θ = (-7/25) / (-24/25)
tan θ = (-7/25) * (-25/24)
tan θ = 7/24
Therefore, sin θ = -7/25 and tan θ = 7/24.