Unit 5, Lesson 5

Six Trigonometric Ratios for Sides of a Right Triangle for Any Acute Angle Theta

Sin = opposite/hypotenuse Cos = adjacent/hypotenuse Tan = opposite/adjacent Csc = hypotenuse/opposite Sec = hypotenuse/adjacent Cot = adjacent/opposite

Inverse trigonometric relations can be used to

find the measure of angles of right triangles (calculators can be used to find values of the inverse trigonometric relations)

arcosine relation

the inverse of the cosine function

arcsine relation

the inverse of the sine function; sin x = sqrt(3)/2 can be written as x = arcsin sqrt(3)/2, which is read "x is an angle whose sine is sqrt(3)/2" or "x equals the arcsine of sqrt(3)/2"; solution, x, consists of all angles that have sqrt(3)/2 as the value of sine x

arctangent relation

the inverse of the tangent function

If theta is one of the acute angles, then

the legs of the triangle are designated as either opposite or adjacent in relation to the angle theta

to solve a triangle

to find all of the measures of its sides and angles (usually, two measures are given; then you can find the remaining measures)

To find the length of a side if you know an angle degree and the length of a side,

use the appropriate trigonometric function and simplify to solve for the length of the side

To find the degree of an angle if you know the lengths of the sides,

use the inverse of the trigonometric function and multiply it by the ratio formed by the lengths of the sides (make sure your calculator is in degree mode)

The hypotenuse length is determined by

using the Pythagorean Theorem (a^2 + b^2 = c^2 where a & b are lengths of the legs and c is the length of the hypotenuse)

Inverses of the Trigonometric Functions

y = sin x --> x = sin^-1y or x = arcsin y y = cos x --> x = cos^-1y or x = arccos y y = tan x --> x = tan^-1y or x = arctan y