Find the exact value of the remaining trigonometric functions of 𝜃 given that cos 𝜃 = ⁡ −⁡5/9 ⁡⁡⁡⁡,⁡⁡⁡⁡⁡⁡⁡⁡𝜋⁡ ≤ ⁡𝜃⁡ ≤ ⁡ 3𝜋 2

Answer 1

Hello,

Here you just have to remember that the trigonometric functions are all just the ratios of different sides of a triangle. This means that you can use the known values to build up a triangle which can then be used to find the other trigonometric functions.

For example,

If sin(θ) = (√3)/2

and sine = opp/hyp

Then we know that the opposite side of the triangle is √3 units long and the hypotenuse is 2 units long.

From there we can use the Pythagorean theorem to solve for the adjacent side.

a² + b² = c²

we know

a = (√3)/2

c = 1

b = adj

[(√3)/2]² + b² = 1²

Now solve for b

b = √(1 - 3/4)

b = 1/2

Now we know all the sides.

Your question is also telling you that your triangle sits in the 3rd quadrant, meaning both x and y are negative. Let's assume sin(θ) = (√3)/2 is in the 3rd quadrant as well.

To find the other trig functions,

y = opp = -(√3)/2

x = adj = -1/2

hyp = 1

sin(θ) = -(√3)/2

cos(θ) = -1/2

tan(θ) = √3

csc(θ) = -2/√3

sec(θ) = -2

cot(θ) = 1/√3

Let me know if this helped or if you need any further clarification!