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TRIGONOMETRYLesson 2: Solving Right Triangles
Todays Objectives
• Students will be able to develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles, including:• Solve right triangles, with or without technology
Using Trigonometry to solve for a side
• In order to find an unknown side measure in a right triangle using trig ratios, the length of one other side and the measure of one of the acute angles is required.
• Example: Solve for the side length x to the nearest tenth of a centimeter
• Solution: First, identify the positions of the side lengths relative to the acute angle whose measure is known
• Since the length of the hypotenuse and opposite side is required (side x), the opposite-hypotenuse ratio is used. This is the sine ratio
10 cm62°
x
hypotenuseadjacent
opposite
Example• Solve for side length x to the nearest tenth of a meter
• Solution:
46 m
x
37°
Example• Triangle ABC has ےC = 90º, ےA = 55º, and AC = 50 in.
Solve for side length AB to the nearest inch.• Solution: First, draw and label a sketch of a representative
triangle.
• Since the length of the adjacent side relative to ےA is known (50 in) and the length of the hypotenuse (AB) is required, the adjacent-hypotenuse ratio is used. This is the cosine ratio.
A
B
C55° 50 in.
x
Using Trigonometry to Solve for an Angle
• In order to find the measure of one of the acute angles in a right triangle when the measure of each acute angle is unknown, the lengths of two of the three sides must be known
• Solve for to the nearest tenth of a degree
𝜃
15 cm
6 cm
Example• Solution: Label the given sides relative to the angle whose
measure you are trying to determine.
• Since the lengths of the opposite side and the hypotenuse are known, the opposite-hypotenuse ratio is used. This is the sine ratio.
• Use the inverse since function, sin-1, to solve for
𝜃
hypotenuseopposite
adjacent
Example (You do)• Solve for to the nearest tenth of a degree.
• Solution: Since the lengths of the adjacent side and opposite side are known, apply the tangent ratio.
• Use the inverse tan function, tan-1, to solve for
𝜃
30 m
40 m
Solving Triangles
• Often, you will need to determine all the unknown measures of the sides and angles of a triangle. This is referred to as solving the triangle.
• In order to solve a triangle, it is common to use one or more of the following:• Pythagorean theorem• Sum of angles in a triangle• Trigonometric ratios
Example• Solve the following triangle. Give side lengths to the
nearest tenth of a centimeter.
• Solution: If it has not already been done for you, label the unknowns that you are to find
• The measure of angle can be found first
𝜃12 cm y
x 35°
Example• Next, the length of the side labeled x can be determined
using a trig ratio.
• The final side, y, can be determined either using the Pythagorean theorem or a trig ratio. Let’s use the trig ratio, sine.
Example (You do)• Solve the following triangle. Give the unknown side length to
the nearest tenth of a centimeter, and give unknown angles to the nearest tenth of a degree.
• Solution: Side AC = 6 cm, Angle A = 53.1°, Angle B = 36.9°
C B
A
8 cm
10 cm
Homework• Exercises #3-16, pg. 111-112• Begin to add the chapter 2 vocabulary words to your
vocabulary books. These are the words in BOLD in your textbook (the first one is angle of inclination….the second one is tangent ratio, etc.)