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.)