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Math II ยท G-SRT.8
Using Trig Ratios and the Pythagorean Theorem to Solve Right Triangles in Applied Problems
Right-triangle trigonometry turns angles and distances into usable measurements when direct measurement is inconvenient, impossible, or unsafe.
Concept
Geometry
Domain
Similarity, Right Triangles, and Trigonometry
Read time
1 minutes
Problem Library
Problems in the App From This Objective
180 problems across 15 archetypes in the app.
identify legs/hypotenuse and solve.
Problem 1
Find the missing side using the Pythagorean Theorem from legs 6 and 8.
Problem 2
Find the missing side using the Pythagorean Theorem from hypotenuse 13 and leg 5.
Problem 3
Find the missing side using the Pythagorean Theorem from legs 7 and 9.
Problem 4
Find the missing side using the Pythagorean Theorem from hypotenuse 10 and leg 6.
Problem 5
Problem 6
Find the missing side using the Pythagorean Theorem from legs 3 and 4.
Find the missing side using the Pythagorean Theorem from hypotenuse 17 and leg 8.
Open in simulator
Problem 7
Find the missing side using the Pythagorean Theorem from legs 5 and 6.
Problem 8
Find the missing side using the Pythagorean Theorem from hypotenuse 10 and leg 7.
Problem 9
Find the missing side using the Pythagorean Theorem from legs 5 and 12.
Problem 10
Find the missing side using the Pythagorean Theorem from hypotenuse 25 and leg 7.
Problem 11
Find the missing side using the Pythagorean Theorem from legs 2 and 4.
Problem 12
Find the missing side using the Pythagorean Theorem from hypotenuse 8 and leg 4.
set up opposite/hypotenuse ratio.
Problem 13
Find the missing side using sine from angle 30 degrees, hypotenuse 10, opposite x.
Problem 14
Find the missing side using sine from angle theta, hypotenuse h, opposite x.
Problem 15
Find the missing side using sine from angle 45 degrees, opposite 7, hypotenuse h.
Problem 16
Find the missing side using sine from angle 60 degrees, hypotenuse 12, opposite x.
Problem 17
Find the missing side using sine from angle 30 degrees, opposite 6, hypotenuse h.
Problem 18
Find the missing side using sine from angle 45 degrees, hypotenuse 10, opposite x.
Problem 19
Problem 20
Find the missing side using sine from angle 60 degrees, opposite 9, hypotenuse h.
Find the missing side using sine from angle alpha, hypotenuse k, opposite y.
Open in simulator
Problem 21
Find the missing side using sine from angle 30 degrees, hypotenuse 14, opposite y.
Problem 22
Find the missing side using sine from angle 45 degrees, opposite 5, hypotenuse z.
Problem 23
Find the missing side using sine from angle 60 degrees, hypotenuse 8, opposite m.
Problem 24
Find the missing side using sine from angle beta, opposite p, hypotenuse q.
set up adjacent/hypotenuse ratio.
Problem 25
Problem 26
Find the missing side using cosine from angle 60 degrees, hypotenuse 12, adjacent x.
Find the missing side using cosine from angle theta, hypotenuse h, adjacent x.
Open in simulator
Problem 27
Find the missing side using cosine from angle 30 degrees, adjacent 8, hypotenuse h.
Problem 28
Find the missing side using cosine from angle 45 degrees, hypotenuse 10, adjacent x.
Problem 29
Find the missing side using cosine from angle 30 degrees, hypotenuse 10, adjacent x.
Problem 30
Find the missing side using cosine from angle 60 degrees, adjacent 5, hypotenuse h.
Problem 31
Find the missing side using cosine from angle 20 degrees, hypotenuse 15, adjacent y.
Problem 32
Find the missing side using cosine from angle 70 degrees, adjacent 7, hypotenuse z.
Problem 33
Find the missing side using cosine from angle alpha, hypotenuse k, adjacent m.
Problem 34
Find the missing side using cosine from angle beta, adjacent p, hypotenuse q.
Problem 35
Find the missing side using cosine from angle pi/3 radians, hypotenuse 8, adjacent x.
Problem 36
Find the missing side using cosine from angle pi/4 radians, adjacent 6, hypotenuse h.
set up opposite/adjacent ratio.
Problem 37
Find the missing side using tangent from angle 45 degrees, adjacent 7, opposite x.
Problem 38
Find the missing side using tangent from angle theta, adjacent a, opposite x.
Problem 39
Find the missing side using tangent from angle 30 degrees, opposite 5, adjacent x.
Problem 40
Find the missing side using tangent from angle 60 degrees, adjacent 4, opposite x.
Problem 41
Find the missing side using tangent from angle 45 degrees, opposite 10, adjacent x.
Problem 42
Find the missing side using tangent from angle 30 degrees, adjacent 6, opposite x.
Problem 43
Find the missing side using tangent from angle 60 degrees, opposite 12, adjacent x.
Problem 44
Problem 45
Find the missing side using tangent from angle alpha, opposite b, adjacent x.
Find the missing side using tangent from angle 45 degrees, adjacent 9, opposite y.
Open in simulator
Problem 46
Find the missing side using tangent from angle 30 degrees, opposite 7, adjacent y.
Problem 47
Find the missing side using tangent from angle 60 degrees, adjacent 5, opposite y.
Problem 48
Find the missing side using tangent from angle beta, adjacent c, opposite y.
use ratio and inverse trig.
Problem 49
Find the missing acute angle using inverse sine from opposite 5, hypotenuse 10.
Problem 50
Find the missing acute angle using inverse sine from opposite 3, hypotenuse 5.
Problem 51
Find the missing acute angle using inverse sine from opposite 8, hypotenuse 17.
Problem 52
Find the missing acute angle using inverse sine from opposite o, hypotenuse h.
Problem 53
Find the missing acute angle using inverse sine from opposite 4, hypotenuse 5.
Problem 54
Find the missing acute angle using inverse sine from opposite 7, hypotenuse 25.
Problem 55
Find the missing acute angle using inverse sine from opposite 12, hypotenuse 13.
Problem 56
Problem 57
Find the missing acute angle using inverse sine from opposite 1, hypotenuse 4.
Find the missing acute angle using inverse sine from opposite 2, hypotenuse 3.
Open in simulator
Problem 58
Find the missing acute angle using inverse sine from opposite 5, hypotenuse 13.
Problem 59
Find the missing acute angle using inverse sine from opposite 15, hypotenuse 17.
Problem 60
Find the missing acute angle using inverse sine from opposite 1, hypotenuse 10.
use ratio and inverse trig.
Problem 61
Find the missing acute angle using inverse cosine from adjacent 6, hypotenuse 12.
Open in simulator
Problem 62
Find the missing acute angle using inverse cosine from adjacent 4, hypotenuse 5.
Problem 63
Find the missing acute angle using inverse cosine from adjacent 15, hypotenuse 17.
Problem 64
Find the missing acute angle using inverse cosine from adjacent a, hypotenuse h.
Problem 65
Find the missing acute angle using inverse cosine from adjacent 12, hypotenuse 13.
Problem 66
Find the missing acute angle using inverse cosine from adjacent 24, hypotenuse 25.
Problem 67
Find the missing acute angle using inverse cosine from adjacent 3, hypotenuse 7.
Problem 68
Find the missing acute angle using inverse cosine from adjacent 1, hypotenuse 3.
Problem 69
Find the missing acute angle using inverse cosine from adjacent 9, hypotenuse 10.
Problem 70
Find the missing acute angle using inverse cosine from adjacent 8, hypotenuse 17.
Problem 71
Find the missing acute angle using inverse cosine from adjacent 2, hypotenuse 3.
Problem 72
Find the missing acute angle using inverse cosine from adjacent 7, hypotenuse 10.
use ratio and inverse trig.
Problem 73
Find the missing acute angle using inverse tangent from opposite 7, adjacent 7.
Problem 74
Find the missing acute angle using inverse tangent from opposite 3, adjacent 4.
Problem 75
Find the missing acute angle using inverse tangent from opposite 8, adjacent 15.
Problem 76
Problem 77
Find the missing acute angle using inverse tangent from opposite o, adjacent a.
Find the missing acute angle using inverse tangent from opposite 10, adjacent 10.
Open in simulator
Problem 78
Find the missing acute angle using inverse tangent from opposite 1, adjacent sqrt(3).
Problem 79
Find the missing acute angle using inverse tangent from opposite sqrt(3), adjacent 1.
Problem 80
Find the missing acute angle using inverse tangent from opposite 5, adjacent 12.
Problem 81
Find the missing acute angle using inverse tangent from opposite 9, adjacent 40.
Problem 82
Find the missing acute angle using inverse tangent from opposite 20, adjacent 21.
Problem 83
Find the missing acute angle using inverse tangent from opposite 1, adjacent 5.
Problem 84
Find the missing acute angle using inverse tangent from opposite 4, adjacent 3.
find all missing sides and angles.
Problem 85
Solve the right triangle from one acute angle is 30 degrees and hypotenuse is 10.
Problem 86
Solve the right triangle from legs are 6 and 8.
Problem 87
Solve the right triangle from acute angle is 45 degrees and one leg is 7.
Problem 88
Solve the right triangle from acute angle theta and hypotenuse h are given.
Problem 89
Solve the right triangle from acute angle 60 degrees, adjacent leg 4.
Problem 90
Solve the right triangle from acute angle 30 degrees, opposite leg 5.
Problem 91
Solve the right triangle from one leg 5, hypotenuse 13.
Problem 92
Solve the right triangle from one leg 3, hypotenuse 6.
Problem 93
Solve the right triangle from acute angle 60 degrees, hypotenuse 12.
Problem 94
Solve the right triangle from legs are 5 and 12.
Problem 95
Problem 96
Solve the right triangle from acute angle is 60 degrees and the leg opposite to it is 9.
Solve the right triangle from acute angle is 30 degrees and the leg adjacent to it is 9.
Open in simulator
model right triangle and choose ratio.
Problem 97
Solve the angle of elevation or depression problem: A tree is 30 ft tall and its shadow is 40 ft long.
Problem 98
Solve the angle of elevation or depression problem: A person 100 m from a tower sees the top at 25 degrees.
Problem 99
Solve the angle of elevation or depression problem: From a cliff 60 m high, the angle of depression to a boat is 12 degrees.
Problem 100
Solve the angle of elevation or depression problem: A kite string is 80 ft and makes a 50 degree angle with the ground.
Problem 101
Solve the angle of elevation or depression problem: A ladder 15 ft long leans against a wall, reaching a height of 12 ft.
Problem 102
Problem 103
Solve the angle of elevation or depression problem: An airplane at an altitude of 5000 ft is 8000 ft horizontally from the airport.
Solve the angle of elevation or depression problem: A plane takes off at an angle of 15 degrees. After flying 2000 m along its path, what is its horizontal distance from the takeoff point?.
Open in simulator
Problem 104
Solve the angle of elevation or depression problem: From a point 50 meters away from the base of a building, the angle of elevation to the top is 40 degrees.
Problem 105
Solve the angle of elevation or depression problem: A flagpole is 25 ft tall. The angle of elevation from a point on the ground to its top is 35 degrees. What is the distance from that point to the top of the flagpole?.
Problem 106
Solve the angle of elevation or depression problem: A boat is 500 m from the base of a lighthouse. The line of sight from the top of the lighthouse to the boat is 600 m.
Problem 107
Solve the angle of elevation or depression problem: A person on a bridge observes a boat below. The angle of depression is 20 degrees, and the boat is 100 ft horizontally from the point directly below the observer.
Problem 108
Solve the angle of elevation or depression problem: A hot air balloon is 300 m high. From a point on the ground, the angle of elevation to the balloon is 60 degrees.
use trig or Pythagorean relationship.
Problem 109
Solve the ladder, ramp, or support problem: A 13-ft ladder reaches 12 ft up a wall.
Problem 110
Solve the ladder, ramp, or support problem: A ramp rises 3 ft over a horizontal run of 12 ft.
Problem 111
Solve the ladder, ramp, or support problem: A support cable makes a 60 degree angle with the ground and reaches 20 ft high.
Problem 112
Solve the ladder, ramp, or support problem: A ladder makes a 75 degree angle with the ground and is 16 ft long.
Problem 113
Solve the ladder, ramp, or support problem: A ladder is placed 6 ft from a wall and reaches 8 ft up the wall.
Problem 114
Solve the ladder, ramp, or support problem: A 10-ft ramp rises 5 ft vertically.
Problem 115
Solve the ladder, ramp, or support problem: A 30-ft support cable makes a 45-degree angle with the ground.
Problem 116
Problem 117
Solve the ladder, ramp, or support problem: A 20-ft ladder makes a 30-degree angle with the wall.
Solve the ladder, ramp, or support problem: A ramp has a horizontal run of 15 ft and makes a 10-degree angle with the ground.
Open in simulator
Problem 118
Solve the ladder, ramp, or support problem: A support cable is attached 10 ft up a pole and makes a 20-degree angle with the pole.
Problem 119
Solve the ladder, ramp, or support problem: A ladder reaches 9 ft up a wall and makes a 70-degree angle with the ground.
Problem 120
Solve the ladder, ramp, or support problem: A 25-ft ramp has a horizontal run of 20 ft.
decompose distance and angle relationships.
Problem 121
Solve the navigation or bearing-style right-triangle problem: A boat travels 12 km east and then 5 km north.
Problem 122
Problem 123
Solve the navigation or bearing-style right-triangle problem: A hiker walks 8 mi north then 6 mi west.
Solve the navigation or bearing-style right-triangle problem: A plane flies 100 miles at 30 degrees north of east.
Open in simulator
Problem 124
Solve the navigation or bearing-style right-triangle problem: A ship is 20 km from port at bearing forming a 40 degree angle east of north.
Problem 125
Solve the navigation or bearing-style right-triangle problem: A bird flies 7 km south and then 24 km west.
Problem 126
Solve the navigation or bearing-style right-triangle problem: A car drives 90 km east and then 120 km south.
Problem 127
Solve the navigation or bearing-style right-triangle problem: A submarine travels 50 miles at 60 degrees south of west.
Problem 128
Solve the navigation or bearing-style right-triangle problem: An airplane flies 200 km at 25 degrees west of south.
Problem 129
Solve the navigation or bearing-style right-triangle problem: A person walks 15 meters west and then 8 meters north.
Problem 130
Solve the navigation or bearing-style right-triangle problem: A drone flies 20 km south and then 21 km east.
Problem 131
Solve the navigation or bearing-style right-triangle problem: A boat sails 75 nautical miles at 45 degrees north of west.
Problem 132
Solve the navigation or bearing-style right-triangle problem: A tracking device indicates an object is 150 meters away at 35 degrees east of south.
choose method from known information.
Problem 133
Decide whether to use a trig ratio or the Pythagorean Theorem for two side lengths are known and one side is unknown.
Open in simulator
Problem 134
Decide whether to use a trig ratio or the Pythagorean Theorem for one acute angle and one side are known and another side is unknown.
Problem 135
Decide whether to use a trig ratio or the Pythagorean Theorem for two sides are known and an acute angle is unknown.
Problem 136
Decide whether to use a trig ratio or the Pythagorean Theorem for only the two acute angles are known.
Problem 137
Decide whether to use a trig ratio or the Pythagorean Theorem for the lengths of the two legs are known and the hypotenuse is unknown.
Problem 138
Decide whether to use a trig ratio or the Pythagorean Theorem for the hypotenuse and one leg are known and the other leg is unknown.
Problem 139
Decide whether to use a trig ratio or the Pythagorean Theorem for an acute angle and the hypotenuse are known and the opposite leg is unknown.
Problem 140
Decide whether to use a trig ratio or the Pythagorean Theorem for an acute angle and an adjacent leg are known and the hypotenuse is unknown.
Problem 141
Decide whether to use a trig ratio or the Pythagorean Theorem for the lengths of the opposite leg and hypotenuse are known and an acute angle is unknown.
Problem 142
Decide whether to use a trig ratio or the Pythagorean Theorem for the lengths of the adjacent leg and hypotenuse are known and an acute angle is unknown.
Problem 143
Decide whether to use a trig ratio or the Pythagorean Theorem for only one side length is known and another side is unknown.
Problem 144
Decide whether to use a trig ratio or the Pythagorean Theorem for only one side length is known and an acute angle is unknown.
use context and measurement precision.
Problem 145
Round the right-triangle answer appropriately for height calculated as 17.462 ft, nearest tenth requested.
Problem 146
Round the right-triangle answer appropriately for angle calculated as 36.869897 degrees, nearest degree requested.
Problem 147
Round the right-triangle answer appropriately for distance calculated as 42.04 m, nearest meter requested.
Problem 148
Round the right-triangle answer appropriately for exact length 6sqrt(3) with exact form requested.
Problem 149
Round the right-triangle answer appropriately for length calculated as 5.1234 cm, nearest hundredth requested.
Problem 150
Problem 151
Round the right-triangle answer appropriately for angle calculated as 22.567 degrees, nearest tenth of a degree requested.
Round the right-triangle answer appropriately for area calculated as 123.45 square inches, nearest square inch requested.
Open in simulator
Problem 152
Round the right-triangle answer appropriately for hypotenuse calculated as 15.00045 units, nearest thousandth requested.
Problem 153
Round the right-triangle answer appropriately for side length calculated as 8.99 feet, nearest foot requested.
Problem 154
Round the right-triangle answer appropriately for angle calculated as 45.33 degrees, nearest degree requested.
Problem 155
Round the right-triangle answer appropriately for exact length 5sqrt(2) with exact form requested.
Problem 156
Round the right-triangle answer appropriately for perimeter calculated as 30.123 cm, nearest tenth requested.
state what side/angle represents.
Problem 157
Interpret the right-triangle solution in context: x=24 ft in a tree-shadow problem.
Problem 158
Interpret the right-triangle solution in context: theta=38 degrees for a ramp.
Problem 159
Interpret the right-triangle solution in context: d=120 m in an angle of depression problem.
Problem 160
Interpret the right-triangle solution in context: h=15 ft for a ladder problem.
Problem 161
Interpret the right-triangle solution in context: alpha=60 degrees for a kite string.
Problem 162
Interpret the right-triangle solution in context: w=50 meters for a river width.
Problem 163
Interpret the right-triangle solution in context: H=75 meters for a building height.
Problem 164
Interpret the right-triangle solution in context: L=30 feet for a guy wire.
Problem 165
Interpret the right-triangle solution in context: beta=25 degrees for a boat from a lighthouse.
Problem 166
Problem 167
Interpret the right-triangle solution in context: D=500 km for a plane's horizontal distance.
Interpret the right-triangle solution in context: f=40 feet for a flagpole.
Open in simulator
Problem 168
Interpret the right-triangle solution in context: gamma=45 degrees for a playground slide.
catch wrong ratio, wrong reference angle, hypotenuse confusion, and rounding mistakes.
Problem 169
Correct the right-triangle application error: A student uses cosine for opposite over hypotenuse.
Problem 170
Correct the right-triangle application error: A student treats the ladder length as the vertical height.
Problem 171
Correct the right-triangle application error: A student uses the angle at the top but labels sides from the ground angle.
Problem 172
Correct the right-triangle application error: A student rounds each intermediate value to the nearest whole number.
Problem 173
Correct the right-triangle application error: A student uses tangent to find the hypotenuse when given the opposite side and an angle.
Problem 174
Correct the right-triangle application error: A student uses the ratio of adjacent over opposite for the tangent function.
Problem 175
Correct the right-triangle application error: A student uses the sine function to calculate an angle measure.
Problem 176
Problem 177
Correct the right-triangle application error: A student labels the side adjacent to the reference angle as the opposite side.
Correct the right-triangle application error: A student incorrectly assumes the side opposite the given angle is the hypotenuse.
Open in simulator
Problem 178
Correct the right-triangle application error: A student calculates trigonometric values in radians instead of degrees.
Problem 179
Correct the right-triangle application error: A student uses the right angle (90 degrees) as the reference angle for sine or cosine.
Problem 180
Correct the right-triangle application error: A student calculates the angle of depression as an angle above the horizontal line of sight.