Answer:
sin(θ) = √65/9
tan(θ) = √65/4
sec(θ) = 9/4
csc(θ) = 9/√65 = (9√65)/65
cot(θ) = 4/√65 = (4√65)/65
Step-by-step explanation:
Let's first define all these trigonometric functions:
- sine (sin) = opposite / hypotenuse
- cosine (cos) = adjacent / hypotenuse
- tangent (tan) = opposite / adjacent
- secant (sec) = hypotenuse / adjacent
- cosecant (csc) = hypotenuse / opposite
- cotangent (cot) = adjacent / opposite
We know that cos(θ) = 4/9, where 4 is the adjacent and 9 is the hypotenuse. We can make this into a right triangle (see attachment), where one of the legs is 4, the hypotenuse is 9, and the last leg is √(9² - 4²) = √(81 - 16) = √65. That means the opposite is √65.
We can now solve for all the rest of the trigonometric functions:
sin(θ) = √65/9
tan(θ) = √65/4
sec(θ) = 9/4
csc(θ) = 9/√65 = (9√65)/65
cot(θ) = 4/√65 = (4√65)/65
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