GPS Pre-Calculus Keeper 10 Right Triangle Trig GPS Pre-Calculus Keeper 10

Trigonometric Ratios The lengths of the sides of a right triangle are used to define the six trigonometric ratios: sine (sin) cosine (cos) tangent (tan) cosecant (csc) secant (sec) cotangent (cot) Side opposite  Side adjacent to  Hypotenuse 

Trigonometric Function Values of an Acute Angle  Let  be an acute angle of a right triangle. Then the six trigonometric functions of  are as follows:

Memorizing Trigonometric Ratios The mnemonic device SOH – CAH – TOA Is most commonly used to remember the ratios for sine, cosine, and tangent.

Reciprocal Functions Note that there is a reciprocal relationship between pairs of the trigonometric functions.

Common Relationships From the definitions of the sine, cosine, tangent, and cotangent functions, you can also derive the following relationships.  

Example In the triangle shown, find the six trigonometric function values of (a)  and (b) .   13 5

Example If and  is an acute angle, find the other five trigonometric function values of .

Find the value of x. Round to the nearest tenth.

Application A competitor in a triathlon is running along the course shown. Determine the length in feet that the runner must cover to reach the finish line.

Investigation of Special Right Triangle

Trigonometric Values of Special Angles

Inverse Trigonometric Functions  

Find a Missing Angle Measure  

Angle of Elevation and Depression Some applications of trig use an angle of elevation or depression. An angle of elevation is the angle formed by a horizontal line and an observer’s line of sight to an object above. An angle of depression is the angle formed by a horizontal line and an observer’s line of sight to an object below.

Using an Angle of Elevation A ground crew worker who is 6 feet tall is directing a plane on a runway. If the worker sights the plane at an angle of elevation of 32 degrees, what is the horizontal distance from the worker to the plane?

Application A hot air balloon that is moving above a neighborhood has an angle of depression of 28 degrees to one hose and 52 degrees to a house down the street. If the height of the balloon is 650 feet, estimate the distance between the two houses.

Solving Right Triangles Trigonometric functions and inverse relations can be used to solve a right triangle, which means to find the measures of all of the sides and angles of the triangle.