length of hypotenuse

length of leg opposite

length of hypotenuselength of leg adjacent to

7.6 Apply the Sine and Cosine Ratios Goal Use the sine and cosine ratios.

SINE AND COSINE RATIOSLet ΔABC be a right triangle with acute A. The sine of A and cosine of A (written sin A and cos A) are defined as follows:

sin A = =

cos A = =

Example: Find sin U and sin W. Write each answer as a fraction.

sin U=opphyp

= sin W=opphyp

=

Example: Find cos S and cos R. Write each answer as a fraction

cos S= cosR=

Example: Basketball You walk from one corner of a basketball court to the opposite corner. Write and solve a proportion using a trigonometric ratio to approximate the distance of the walk.

Step1: Determine which two sides are involved for the given ∠. (in this case opp and hyp)

Step2: Choose the appropriate trig function (in this case opp and hyp -> sin)

Step3: Set up and solve a proportion. sin 62°

1=94x

Remember these abbreviations:sine → sincosine → coshypotenuse → hyp

Angle of depression - When looking down at an object, the angle your line of sight makes with a horizontal line is called the angle of depression.

Example: Roller Coaster You are at the top of a roller coaster 100 feet above the ground. The angle of depression is 44°. About how far do you ride down the hill?

Step1:

Step2:

Step3:

Angle of elevation - When looking up at an object, the angle your line of sight makes with a horizontal line is called the angle of elevation.

Example: Railroad A railroad crossing arm that is 20 feet long is stuck with an angle of elevation of 35° Find the lengths x and y

Examples: Find the value of each variable. Round decimals to the nearest tenth.

What if you have the sides and want to find the angle? (AKA what undoes sine and cosine?)(AKA What the heck is that sin−1∧cos−1 thing on my calculator?)

Step1: Identify sides for ∡ A You Try (Find m∡ A):

Step2: Set up appropriate ratio

Step3: “Undo” sin or cos with sin−1∨cos−1on calculator