6.2 Trigonometric Applications Objectives: Solve triangles using trigonometric ratios. Solve applications using triangles.

Ex. #1 Find a Side of a Triangle Find the side x of the right triangle given below: SOH-CAH-TOA Since they two sides being used on this problem are the adjacent & the hypotenuse, then cosine will be used. Make sure your calculator is in degree mode. hyp opp adj Ex. #1 Find a Side of a Triangle

Ex. #1 Find a Side of a Triangle Find the side x of the right triangle given below: Since the opposite and adjacent are being used, tangent is chosen. SOH-CAH-TOA hyp opp adj Ex. #1 Find a Side of a Triangle

Ex. #1 Find a Side of a Triangle Find the side x of the right triangle given below: SOH-CAH-TOA Since the opposite and hypotenuse are being used, sine is chosen. hyp opp adj Ex. #1 Find a Side of a Triangle

Ex. #2 Find an Angle of a Triangle Find the measure of angle θ in the triangle below: SOH-CAH-TOA Since all sides are given, ANY trig ratio can be used to solve the problem. hyp adj opp Ex. #2 Find an Angle of a Triangle

Ex. #2 Find an Angle of a Triangle Find the measure of angle θ in the triangle below: SOH-CAH-TOA Since the adjacent and hypotenuse are given, cosine will be used. hyp opp adj Ex. #2 Find an Angle of a Triangle

Ex. #2 Find an Angle of a Triangle Find the measure of angle θ in the triangle below: SOH-CAH-TOA Since the opposite & adjacent are given, tangent is chosen. opp adj hyp Ex. #2 Find an Angle of a Triangle

Ex. #3 Solving a Right Triangle Solve the right triangle shown below: SOH-CAH-TOA There are 3 things to solve for on this problem. Sides a and b, & angle θ. To find θ we simply subtract the other two angles from 180°. opp θ = 180° − 90° − 20° = 70° hyp adj Ex. #3 Solving a Right Triangle

Ex. #3 Solving a Right Triangle Solve the right triangle shown below: SOH-CAH-TOA To find a, we will use the opposite and the hypotenuse, so sine is chosen. To find b, we will use the adjacent and the hypotenuse, so cosine is chosen. opp hyp adj Ex. #3 Solving a Right Triangle

Ex. #4 Solving a Right Triangle Solve the right triangle shown below: SOH-CAH-TOA Since the two legs are the same length, then this triangle is isosceles and the angles of θ and β are congruent and equal to 45°. hyp There are four easy ways to find the hypotenuse c: Trig with sine Trig with cosine Pythagorean Theorem 45°-45°-90° Special Right Triangle Rule adj opp Ex. #4 Solving a Right Triangle

Ex. #4 Solving a Right Triangle Solve the right triangle shown below: SOH-CAH-TOA hyp adj opp Ex. #4 Solving a Right Triangle

A wheelchair ramp is 6 feet in length and makes a 4° angle with the ground. How many inches does the ramp rise off the ground? hyp 4° x 6 opp adj Since the opposite and hypotenuse are being used, sine will be chosen. Ex. #5 Application

A diagonal path through a rectangular park is 600 ft. long A diagonal path through a rectangular park is 600 ft. long. One side of the park measures 350 ft. long. How long is the other side of the park? What angle does the diagonal path make with the side you found in question A? 600 350 x θ Ex. #6 Application

Ex. #7 Angle of Elevation & Depression The angle of elevation from a point on the street to the top of a building is 53°. The building is 60 ft. high. How far is the point on the street from the foot of the building? 53° 60 x Ex. #7 Angle of Elevation & Depression

Ex. #8 Angle of Elevation & Depression From the top of a 60 ft lighthouse, built on a cliff 40 ft. above sea level, the angle of depression to a sailboat adrift on the water is 55°. How far from the base of the cliff is the sailboat? The angle of depression is equal to the angle of elevation. Additionally to form the right triangle we must add the height of the cliff to that of the lighthouse. 40 ft 60 ft x 55° 100 ft 55° Ex. #8 Angle of Elevation & Depression

Ex. #9 Angle of Elevation & Depression While on a nature walk, a person spots a small oak tree with an angle of elevation of 25° to the top of the tree and an angle of depression of 15° to the bottom of the tree from eye level. The eye level is 165 cm. How far is the person standing from the tree? 25° x 15° 165 cm 165 cm Ex. #9 Angle of Elevation & Depression

Ex. #9 Angle of Elevation & Depression While on a nature walk, a person spots a small oak tree with an angle of elevation of 25° to the top of the tree and an angle of depression of 15° to the bottom of the tree from eye level. The eye level is 165 cm. How tall is the tree? 287 + 165 = 452 cm y 25° x 15° 165 cm Ex. #9 Angle of Elevation & Depression