Section 4.3 Right Triangle Trigonometry

Overview In this section we apply the definitions of the six trigonometric functions to right triangles. Before we do that, however, let’s remind ourselves about the Pythagorean Theorem:

A Picture

Example Find the missing side of the right triangle.

4.3 – Right Triangle Trigonometry SOHCAHTOA “Some Old Hippie Came Around Here Trippin’ On Acid.” “Some Old Hog Came Around Here and Took Our Apples

SOH-CAH-TOA The six trigonometric functions of the acute angle are…

4.3 – Right Triangle Trigonometry Find the six trig functions of θ in the triangle below

An Example State the six trigonometric values for angles C and T.

4.3 – Right Triangle Trigonometry Find the sine, cosine, and tangent of 45º using the triangle below. 1 45º 1

Construct a triangle with hypotenuse=1. Find the sine, cosine, and tangent of 45º using your triangle. 1

Finding sides for a 30 – 60 – 90 Triangle Given: Equilaterial triangle of side length 1, and altitude h. We know form geometry that the altitude h, bisects the angle it is drawn from and that it is the perpendicular bisector of the opposite side h 30 ⁰ 60 ⁰ What is the length of the altitude h? 30 ⁰ 60 ⁰

Find the sine, cosine, and tangent of 30º and 60º using the triangle from the last slide. 30º 60º

Summary Deg.Rad.SinCosTan 0º0º 30º 45º1 60º 90º 180º 270º 360º

Cofunctions The sine of an angle is equal to the cosine of its compliment (and vice versa). The tangent of an angle is equal to the cotangent of its compliment (and vice versa). The secant of an angle is equal to the cosecant of its compliment (and vice versa).

4.3 – Right Triangle Trigonometry Cofunctions sin 30º = cos 60º = sin 30º = cos (90º – 30º) tan 57º cot 33º ? cot 33º = tan(90º – 33º) = tan 57º

Solving Right Triangles 1.Write the appropriate trigonometric relationship for the unknown value (there may be more than one). 2.Use your scientific calculator to find the appropriate trigonometric value or angle (make sure your calculator is in degree mode).

Examples

4.3 – Right Triangle Trigonometry The angle of elevation from point X to point Y (above X) is made between the ray XY and a horizontal ray. The angle of depression from point X to point Z (below X) is made between the ray XZ and a horizontal ray. Y a.o.e. X a.o.d. Z

4.3 – Right Triangle Trigonometry At a certain time of day Giant Sam’s shadow is 400 feet long. If the angle of elevation of the sun is 42º, how tall is he? 42º 400 ft. h

h = 400 tan 42º h = 400(.9004) h = ft.