Answer:
- sin(θ) = (2/9)√14; csc(θ) = (9√14)/28
- cos(θ) = 5/9; sec(θ) = 9/5
- tan(θ) = (2/5)√14; cot(θ) = (5√14)/28
Explanation:
Given cos(θ) = 5/9, you want the six trig functions of θ.
Identities
The relevant identities are ...
- sin(θ) = ±√(1 -cos(θ)²)
- tan(θ) = sin(θ)/cos(θ)
- csc(θ) = 1/sin(θ)
- sec(θ) = 1/cos(θ)
- cot(θ) = 1/tan(θ)
Sine
The sine of θ is ...
sin(θ) = √(1 -(5/9)²) = √(81 -25)/9 = (√56)/9
sin(θ) = (2/9)√14
Then the cosecant is ...
csc(θ) = 1/sin(θ) = (9/2)/√14
csc(θ) = (9√14)/28
Tangent
The tangent of θ is ...
tan(θ) = sin(θ)/cos(θ) = ((2/9)√14)/(5/9)
tan(θ) = (2/5)√14
Then the cotangent is ...
cot(θ) = 1/tan(θ) = (5/2)/√14
cot(θ) = (5√14)/28
Secant
The secant of θ is ...
sec(θ) = 1/cos(θ) = 1/(5/9)
sec(θ) = 9/5
The cosine is given in the problem statement.