Trigonomic Ratios
What are the three basic trigonometric ratios?
Sine, Cosine, Tangent
If the opposite is 5 m and the hypotenuse is 13 m, find the angle.
θ=sin^-1(5/13) = 22.6°
What is the angle of elevation?
The angle measured upward from the horizontal to an object.
A ramp is 3 m long and rises 1.5 m. Find the angle of inclination.
sin=1.5/3, =30°
A flagpole is mounted on top of a small building. A person stands 25 m from the building. The angle of elevation to the top of the building is 20°, and to the top of the flagpole is 30°. How tall is the flagpole?
Height of building: 25×tan(20°) = 9.1. Total height: 25×tan(30°) = 14.4 m. Flagpole height = 5.3 m.
If sin=opposite/hypotenuse what is cos?
cos=adjacent/hypotenuse
Find the missing side if hypotenuse = 10 m and angle = 30°.
Opposite=10×sin(30°)=5 m
What is the angle of depression?
The angle measured downward from the horizontal to an object.
A building casts a 12 m shadow. The angle of elevation to the top is 40°. Find the building’s height.
12×tan(40°) = 10.1 m
A hiker is 40 m away from the base of a cliff. The angle of elevation to the top of the cliff is 35°. There’s a tree on top of the cliff. The angle of elevation to the top of the tree is 45°. How tall is the tree?
Cliff height: 40×tan(35°)≈28.040 m. Total height: 40×tan(45°)=4040 \times \tan(45°) = 4040×tan(45°)=40 m. Tree height ≈ 12.0 m.
A triangle has an opposite side of 4 and a hypotenuse of 5. Find the sin ratio.
sin=4/5=0.8
A right triangle has adjacent = 7 cm and angle = 40°. Find the opposite side.
Opposite=7×tan(40°) = 5.9 cm
If the angle of elevation is 25° and distance to base is 10 m, find height.
Height=10×tan(25°)≈4.66 m
A tree is 8 m tall. From a point 5 m away, what is the angle of elevation?
tan^−1(8/5) = 58°
A surveyor stands 60 m away from a tower that sits on a raised platform. The angle of elevation to the top of the platform is 15°, and to the top of the tower is 40°. How tall is the tower itself?
Platform height: ≈16.1 m. Total height: 50.4 m. Tower height ≈ 34.3 m.
If tan=0.75 = what is the ratio of opposite to adjacent?
Opposite : Adjacent = 3 : 4
Opposite = 12 cm, angle = 35°. Find hypotenuse.
Hypotenuse=sin(35°)12 = 20.9 cm
A kite string is 20 m long at an angle of 35°. How high is the kite?
20×sin(35°) = 11.5 m
A person stands 30 m away from a tower. The angle of elevation is 32°. Find the height of the tower.
30×tan(32°) = 18.8 m
A drone hovers above a hill. A person stands 30 m away from the base. The angle of elevation to the top of the hill is 25°, and to the drone is 50°. How high is the drone above the ground?
Hill height = 14.0 m total height 35.8 m drone height = 35.8 m
A right triangle has adjacent = 8 cm and hypotenuse = 10 cm. Find cos ratio.
cos=8/10=0.8
Hypotenuse = 15 cm, adjacent = 9 cm. Find the angle.
cos(θ)=9/15, = 53.1°
A ladder leans against a wall at 60°. If the ladder is 5 m long, how far up the wall does it reach?
5×sin(60°) = 4.33 m
A hot air balloon rises vertically. The angle of elevation from a point on the ground 100 m away is 48°. Find the height.
100×tan(48°) = 111.1 m
A ski jump ramp rises from the ground to a platform. A spectator stands 45 m from the base. The angle of elevation to the platform is 18°, and to the top of a flagpole on the platform is 33°. How tall is the flagpole?
platform height = 14.6 m total height = 29.2 m flagpole height = 14.6 m