Use the given values to evaluate three trigonometric functions cosø, sinø, tanø.

Question

Use the given values to evaluate three trigonometric functions cosø, sinø, tanø.

secø=3/4, cscø= -3√5/5

Answer 1

For the first equation, sec θ = 3/4, the range of sec θ is (-∞,-1]U[1,∞).

Therefore the equation is "invalid", and there exists no such θ that will make it true.

For the second equation, csc θ = -3√5/5

We can use the trig. identity that says csc θ = 1 / sin θ

Therefore,

sin θ = -5 / 3√5 [you can rationalize if you like]

To find cos and tan, we can find the corresponding right triangle and then use the Pythagorean theorem.

Since sin θ = (opposite) / (hypotenuse)

The opposite side length of the triangle will be 5, and the hypotenuse will be 3√5.

Using Pythagoras, 5^2 + X^2 = (3√5)^2, we get X = ±2√5

So, our adjacent side length is ±2√5

Therefore,

cos θ = ±2√5 / 3√5 = ±2/3

tan θ = ±5/2√5 = ±√5 / 2

Answer 2

1. Using the unit circle, since secant is positive and cosecant is negative, ∅ should lie in the IV quadrant on the unit circle.
2. You can figure out Cosine and Sine easily enough by taking the reciprocal of the given Secant and Cosecant (and making sure there are no radicals in the denominator).
3. From there you can use the solved sine and cosine to find Tangent and Cotangent (again, ensuring you have simplified to no radicals in the denominator).
4. So:
5. Sec∅=3/4; Cos∅=4/3 Cosine + in 1st and 4th quadrants only.
6. Ccs∅=-3√5/5; Sin∅=-√5/3 Sine is - in the 3rd and 4th quadrant, therefore, ∅ lies in the 4th quadrant.
7. Tan ∅= sin/cos = -√5/4; Cot∅=-4√5/5

Does this process make sense to you? Do you have any further questions?