Download 14-3A: Right Triangles and Trigonometric Ratios

14-3A: Right Triangles and
Trigonometric Ratios
Objectives:
• To find lengths of missing sides in a right triangle.
• To find measures of missing angles in a right triangle.
Right Triangle Basics:
Labeling Sides and Angles of a Right Triangle
The angles of a triangle and the sides opposite them are named with the same letters.
Angles are named with capital letters and sides are named with lower case letters.
hypotenuse
opposite
adjacent
hypotenuse = the side across from the right angle
adjacent = the side next to one of the smaller angles
opposite = the side across from one of the smaller angles
NOTE: When we are solving for an unknown linear measurement, we use the variable .
When we are solving for an unknown angular measurement, we use the variable .
The Sum of the Angles of a Triangle:
The sum of the angle measures in a triangle is equal to 180o.
EX:
Pythagorean Theorem:
If we have a RIGHT TRIANGLE, we can use the Pythagorean Theorem to find a missing side when given the other two sides.
Remember:
The squares of the two SHORT sides equal the square of the LONGEST side.
EX:
3
4
Practice Problem:
Find the length of the missing side if c = 13 and a = 5.
C
A
B
Find Missing Sides Using Trigonometric Functions:
** In order to use trigonometric functions, you MUST have a RIGHT TRIANGLE! **
S
O
H ­ C
A
H ­ T
O
A
sine:
cosine:
tangent:
NOTE: If you get square roots for the values of adjacent, opposite or hypotenuse, leave them as radicals!!! (Remember to "Get the hippie out of the basement." if necessary.)
EX: Find all three trig functions for the given right triangle.
C
12
B
5
A
13
EX: Given the trig function, find the lengths of all three sides of the right triangle.
D
E
F
EX: Given an angle and one side of a right triangle, find all the missing sides and angles.
In triangle ABC, angle C is the right angle.
The measure of angle B is 20o.
Side a is 6 cm.
B
C
a. Find the length of side c.
b. Find the length of side b.
c. Find the measure of angle A.
A
Checking the MODE on your calculator:
1. Push [MODE]
2. Check to make sure you are in DEGREES (not RADIANS).
Finding Missing Angles using Inverse Trigonometric Functions:
Inverse trig functions allow us to "undo" a trig function so we can find out the measure of a missing angle.
Trig Functions: Inverse Trig Functions:
EX: Solve for angle in each equation.
1.
2.
3.
EX: Given triangle ABC, find the measures indicated.
a. Find the measure of angle A using sine.
17
A
8
15
C
b. Find the measure of angle B using cosine.
B
EX: Given two sides of a right triangle, find the missing angle using the most appropriate trig function.
C
24
B
32
A
EX: Given a trig function, find the measures of the angles of triangle DEF.
D
F
E