Find the remaining trigonometric ratios of θ if sin(θ) is given and θ QI

sin(θ)=7/sqrt 149
cos=
tan=
cot=
csc=
sec=

sinA = 7/sqrt149 = Y/r.

X^2 + Y^2 = r^2,
X^2 + 7^2 = (sqrt149)^2,
X^2 +49 = 149,
X^2 = 100,
X = 10.

X = 10, Y = 7, r = sqrt149.

cosA = X/r = 10 / sqrt149.
tanA = Y/X = 7 / 10.
cotA = X/Y =
cscA =
secA =

To find the remaining trigonometric ratios of θ in Quadrant I given sin(θ) = 7/√149:

1. Start by using the Pythagorean identity to determine the value of cos(θ). In Quadrant I, both sin(θ) and cos(θ) are positive.
- Pythagorean identity: sin^2(θ) + cos^2(θ) = 1
- Since sin(θ) = 7/√149, substitute it into the identity: (7/√149)^2 + cos^2(θ) = 1
- Simplifying, we get: 49/149 + cos^2(θ) = 1
- Rearranging the equation, cos^2(θ) = 1 - 49/149
- Evaluating the expression, we find: cos^2(θ) = 100/149
- Thus, cos(θ) = √(100/149) = 10/√149

2. Next, find tan(θ), using the fact that tan(θ) = sin(θ) / cos(θ):
- tan(θ) = (7/√149) / (10/√149)
- Simplifying, we get: tan(θ) = 7/10

3. To find cot(θ), use the reciprocal of tan(θ):
- cot(θ) = 1 / tan(θ)
- Substituting the value for tan(θ) calculated earlier, we have: cot(θ) = 10/7

4. Similarly, to find csc(θ), use the reciprocal of sin(θ):
- csc(θ) = 1 / sin(θ)
- Substituting the value of sin(θ) given, we get: csc(θ) = √149 / 7

5. Finally, to find sec(θ), use the reciprocal of cos(θ):
- sec(θ) = 1 / cos(θ)
- Substituting the value of cos(θ) calculated, we have: sec(θ) = √149 / 10

So, the remaining trigonometric ratios of θ in Quadrant I are:
cos(θ) = 10/√149
tan(θ) = 7/10
cot(θ) = 10/7
csc(θ) = √149 / 7
sec(θ) = √149 / 10