If sinx=1/3 where pi/2 <= x <= pi, determine the 5 other trigonometric functions
<= means less than or equal to
please help!!!!
the sides of the triangle are 1,√8,3
no plug and chug.
come back if you get stuck, and show where.
To determine the five other trigonometric functions (cosine, tangent, cosecant, secant, and cotangent) given that sin(x) = 1/3, we can use the Pythagorean Identity:
sin^2(x) + cos^2(x) = 1
Substituting sin(x) = 1/3:
(1/3)^2 + cos^2(x) = 1
1/9 + cos^2(x) = 1
cos^2(x) = 1 - 1/9
cos^2(x) = 8/9
Taking the square root of both sides, we get:
cos(x) = ±√(8/9) = ±(2√2/3)
Now, let's determine the other trigonometric functions:
1. Cosine (cos(x)): We have already found two possible values for cos(x): cos(x) = 2√2/3 and cos(x) = -2√2/3.
2. Tangent (tan(x)): To find the tangent, we can use the identity:
tan(x) = sin(x)/cos(x)
Therefore, tan(x) = (1/3) / (2√2/3) = 1 / (2√2) = √2 / 4
3. Cosecant (csc(x)): Cosecant is the reciprocal of sine:
csc(x) = 1/sin(x) = 1/(1/3) = 3
4. Secant (sec(x)): Secant is the reciprocal of cosine:
sec(x) = 1/cos(x)
For cos(x) = 2√2/3, sec(x) = 1 / (2√2/3) = 3 / (2√2)
For cos(x) = -2√2/3, sec(x) = 1 / (-2√2/3) = -3 / (2√2)
5. Cotangent (cot(x)): Cotangent is the reciprocal of tangent:
cot(x) = 1/tan(x) = 1 / (√2 / 4) = 4 / √2 = 2√2
So, the five other trigonometric functions are:
cos(x) = 2√2/3 and cos(x) = -2√2/3
tan(x) = √2/4
csc(x) = 3
sec(x) = 3 / (2√2) and sec(x) = -3 / (2√2)
cot(x) = 2√2