Question

Please help! A right triangle has acute angle theta and cos theta = 2/5. What are the five other ratios of theta?

Answer 1

cos theta = 2/5. Theta is the angle and cos of any angle is the ratio between the Adjacent side of the given triangle and the Hypotenuse side of the same triangle.

Given these values, you can make a quick fraction representing the ratio as A/H, respectively.

With those values, you can determine that for the actual given triangle, the length of the Adjacent side is equal to 2 and the Hypotenuse is equal to 5.

Given this, you can use some quick algebra and the Pythagorean theorem to deduce the value of the 3rd side of the triangle, which we'll aptly call 'Opposite side'.

\[\sqrt{5^{2}-2^{2}} =\sqrt{21}\]

You can now use the value Square Root of 21 or 21^(1/2) as the value of the 'Opposite Side' which we'll shorten to O for the rest of the problem.

sin theta = O/H csc theta = H/O
cos theta = A/H sec theta = H/A
tan theta = O/A cot theta = A/O

You have all the values necessary to plug in and find the values of the ratios for the remaining 5 trigonometric functions(I'm assuming it's these 6 in total) using the ratio given for cos theta.

Answer 2

So for the Sin theta= O/H will equal sqrt21/5?

Answer 3

A/h is 2/5
O/A is sqrt21/2 etc ?

Answer 4

By the way the Pythagorean theorem applies because of the fact that the triangle is a 'RIGHT' triangle.

Also you are unable to actually create a right triangle with an obtuse angle given that even the SUM of the other two angles of a right triangle can't add up to be > 90.

And yes that is the correct values for the given ratios.

Answer 5

Okayy just to make sure all the ratios are these:

\[\sqrt21/5 \]\[2/5\]\[\sqrt 21/2 \]\[5/\sqrt21\]\[5/2\]\[2/\sqrt21\]

Answer 6

Btw thank you so much for this great explanation! Greatly helped!

Answer 7

Yep. You got it. Looks like you're starting to get the hang of trigonometry now~

Answer 8

Thanks to you [: