Given cos(theta)=-4/(sqrt20), 0^@ <= theta <= 180^@cos(θ)=−4√20,0∘≤θ≤180∘
Use the inverse cosine on both sides:
theta = cos^-1(-4/(sqrt20)), 0^@ <= theta <= 180^@θ=cos−1(−4√20),0∘≤θ≤180∘
theta ~~ 153.4^@θ≈153.4∘
You have the value for the cosine function the denominator should be rationalized:
cos(theta)= -4/sqrt20cos(θ)=−4√20
cos(theta)= -4/(2sqrt5)cos(θ)=−42√5
cos(theta)= (-2sqrt5)/5cos(θ)=−2√55
The secant function is the reciprocal of the cosine function:
sec(theta) = 1/cos(theta) sec(θ)=1cos(θ)
sec(theta) = -sqrt5/2sec(θ)=−√52
The sine function can be found using the identity:
sin(theta) = +-sqrt(1-cos^2(theta))sin(θ)=±√1−cos2(θ)
sin(theta) = +-sqrt(1-((-2sqrt5)/5)^2)sin(θ)=±
⎷1−(−2√55)2
sin(theta) = +-sqrt(25/25- 20/25)sin(θ)=±√2525−2025
sin(theta) = +-sqrt5/5sin(θ)=±√55
We know that the sine function is positive in the second quadrant:
sin(theta) = sqrt5/5sin(θ)=√55
The cosecant function is the reciprocal of the sine function:
csc(theta) = 1/sin(theta)csc(θ)=1sin(θ)
csc(theta) = 5/sqrt5csc(θ)=5√5
csc(theta) = sqrt5csc(θ)=√5
Find the tangent function using the identity:
tan(theta) = sin(theta)/cos(theta)tan(θ)=sin(θ)cos(θ)
tan(theta) = (sqrt5/5)/((-2sqrt5)/5)tan(θ)=√55−2√55
tan(theta) = -1/2tan(θ)=−12
The cotangent function is the reciprocal of the tangent function:
cot(theta) = -2cot(θ)=−2
