Trigonometric Ratio: a Ratio of the Lengths of Two Sides of a Right Triangle

GeometryLesson Notes7.4A______

Objective: Find trigonometric ratios in right triangles and find missing sides using trig ratios.

trigonometric ratio: a ratio of the lengths of two sides of a right triangle

The sides of a right triangle are identified relative to an angle in the triangle:

hypotenuse (opposite the right angle)

leg opposite an angle

leg adjacent to an angle (not the hypotenuse)

Practice: Identify hypotenuse and adjacent and opposite sides.

In a right triangle, where A is an acute angle:

sine of A: Ratio of the measure of the leg opposite A to the hypotenuse

sin A =

cosine of A: Ratio of the measure of the leg adjacent to A to the

hypotenuse

cosA =

tangent of A: Ratio of the measure of the leg opposite A to the leg

adjacent to A

cos A =

HELPFUL HINT: Use SOHCAHTOA to remember the trig ratios.

Example 1 (p 365): Find Sine, Cosine, and Tangent Ratios

Find the three trig ratios for L and N.

Express each ratio as a fraction and as a decimal.

Any similar right triangle created using AA Similarity will have proportional sides so the trig ratios will be the same.

The values of trig ratios depends on the measure of the angles.

Scientific and graphing calculators can give the trig ratios for any angle.

Example 2 (p 366): Use a Calculator to Evaluate Expressions

Find each value to the nearest ten thousandth.

BE SURE CALCULATOR IS IN DEGREE MODE!!

tan 56

cos 89

If you know the measures of two sides of a right triangle or the measure of one side and one acute angle of a right triangle, you can use a trig ratio to find all of the pieces of the triangle.

Example 3 (p 366): Use Trigonometric Ratios to Find a Length

a.The hypotenuse of a right triangle measures 12 cm and one acute angle

measures 56. Find the measures of the legs.

b.An acute angle of a right triangle measures 22. The leg opposite the

angle measures 6.7 ft. Find the measures of the other sides.

c.A fitness trainer sets the incline on a treadmill to 7. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor?

 HW A6a p368 #18-21, 28-36

A6b p368 #22-27, 43, 45, 46, 47

A6a/b p368 #18-25, 28-33, 43, 45, 46, 47

GeometryLesson Notes 4.7B______

Objective: Find angle measures in right triangles using inverse trigonometric ratios. Solve

problems involving angle of elevation and angle of depression.

Remember, the values of trig ratios depends on the measure of the angles.

If you know a trig ratio you can use inverse trig ratios to find the measure of the angle.

Scientific and graphing calculators can give the angle associated with any trig ratio.

Example: Use Inverse Trig Ratios to Find Angle Measures

a.cos A = .3346b.sin B = .8091

c.tan D = 4.2315d.cos E = .5

e.cos F = f.tan G =

Example 4 (p 366): Use Trigonometric Ratios to Find an Angle Measure

Find the measures of the acute angles in the triangle.

a. b.

angle of elevation: the angle formed above the horizontal

angle of depression: the angle formed below the horizontal

Example 2 (p 372): Angle of Elevation

A plane takes off from the runway with an angle of elevation of 9. How high will the plane be when it was traveled a horizontal distance of

10,000 feet?

Example 2 (p 372): Angle of Depression

A 10 foot ramp from a moving van to the ground has an angle of

depression of 15. How high is the deck of the moving van above the ground?

 HW A7a p 368 #37-42, 44, 48

p 374 #9, 11, 16-18

A7b fms-Lesson 7.4-7.5 Worksheet

A7c Lesson 7-4 Skills Practice / Practice

fms-Geometry Lesson 7.4Page 1 of 5