Use a sketch to find the exact value of the following expression.

Question

cos (sin^-1 5/13)

 

answer in simplified fraction or integer

Answer 1

All these trig-sketch questions are the same, just figure out what kind of triangle is being talked about, draw it, and decipher what the expression is asking for. The two triangles you absolutely must be familiar with are the 3-4-5 and 5-12-13 triangles!

 

The question is asking for the cos of [ the arcsine of 5/13].

 

Okay, so what is the arcsine of 5/13? It is the angle that has a sine of 5/13. SOHCAHTOA -> Sine is Opposite / Hypotenuse. So 5/13 is Opposite / Hypotenuse. So opposite of the angle in question is a side of length 5, and the hypotenuse is 13. So now you draw that triangle. Because it's a 5-12-13 triangle, the adjacent side must be length 12. So now you go back to the question:

 

• cos of [ the arcsine of 5/13]

 

=

 

• cos of [ the angle we just drew a triangle for]

 

= CAH -> Adjacent / Hypotenuse

 

• 12 / 13