Find values of all six trig functions if sin(theta)= 4/5 and theta is in the second quadrant.
If sin(x)= 4/5 then O=4, H=5 from SOH-CAH-TOA. Using pythagorean thm. yields A=3.
cos(x)=A/H
tan(x)=O/A
csc(x)=H/O
sec(x)=H/A
cot(x)=A/O. In the 2nd quadrant sin,csc are positive, the other 4 are negative.
To find the values of the six trigonometric functions (sine, cosine, tangent, secant, cosecant, and cotangent) when the sine of an angle is given, we can use the following steps:
1. Identify the given information: In this case, the sine of the angle is given as 4/5, and we know that the angle is in the second quadrant.
2. Use the basic trigonometric identity: In the second quadrant, both sine and cosine are positive, while tangent, secant, cosecant, and cotangent are negative.
3. Find the cosine value: Since sine is given as 4/5, we can use the Pythagorean theorem to find the cosine value. The Pythagorean theorem states that for any angle θ in a right triangle, sin^2(θ) + cos^2(θ) = 1. Rearranging the equation, cos^2(θ) = 1 - sin^2(θ). Plugging in the given value, cos^2(θ) = 1 - (4/5)^2 = 1 - 16/25 = 9/25. Taking the square root of both sides, cos(θ) = ± √(9/25) = ± 3/5. Since we are in the second quadrant where the cosine is positive, cos(θ) = 3/5.
4. Find the remaining trigonometric values: Now that we have the sine and cosine values, we can use them to find the remaining trigonometric functions.
- Tangent (tan): tan(θ) = sin(θ) / cos(θ) = (4/5) / (3/5) = 4/3
- Secant (sec): sec(θ) = 1 / cos(θ) = 1 / (3/5) = 5/3
- Cosecant (csc): csc(θ) = 1 / sin(θ) = 1 / (4/5) = 5/4
- Cotangent (cot): cot(θ) = 1 / tan(θ) = 1 / (4/3) = 3/4
Therefore, the values of all six trigonometric functions when sin(θ) = 4/5 and the angle is in the second quadrant are:
sin(θ) = 4/5
cos(θ) = 3/5
tan(θ) = 4/3
sec(θ) = 5/3
csc(θ) = 5/4
cot(θ) = 3/4