5 remaining functions

Question

suppose 0 s an angle in the standard position whose terminal side is in quadrant 1 and sin 0 =84/85. find exact values on the 5 remaining trigonometric functions of 0. i was wrong says has something to do with unit circle. oh I am trying to get this done and correct my work so I may move on to finish my world history.

Answer 1

sin θ = 84 / 85

cos θ = 13 / 85 (85² - 84² = 13²

Can you deter,one the other four functions?

Answer 2

Perhaps it would be easier for you to understand with a picture:

The red triangle is in a circle around the origin. This is not a "unit" circle because a "unit" circle has a radius of 1 and this circle has a radius of 85. We know that because we are given that the sine of θ is 84/85 and we know sine is opposite/hypotenuse so that means the "opposite" is 84 and the hypotenuse (which is also the radius of the circle) is 85. To find out the other side of the triangle (the side marked "x" in the picture), we need to use the Pythagorean Theorem on that red triangle.

x² + 84² = 85²

x² = 85² - 84²

x² = 7225 - 7056

x² = 169

x = √169

x = 13

Now we have all 3 sides of the triangle:

opposite = 84

adjacent = 13

hypotenuse = 85

And we can write all 6 trigonometric ratios:

sin(θ) = opp/hyp = 84/85

cos(θ) = adj/hyp = 13/85

tan(θ) = opp/adj = 84/13

cot(θ) = adj/opp = 13/84

sec(θ) = hyp/adj = 85/13

csc(θ) = hyp/opp = 85/84