Geometry: Three Trig Ratios (SOH CAH TOA) Practice
Practice using sine, cosine, and tangent in right triangles
Geometry: Three Trig Ratios (SOH CAH TOA) Practice
Practice using sine, cosine, and tangent in right triangles
Geometry - Grade 9-12
- 1
In a right triangle, angle A is an acute angle. The side opposite angle A is 7, the side adjacent to angle A is 24, and the hypotenuse is 25. Write sin A, cos A, and tan A as fractions.
Use SOH CAH TOA to match each ratio with the correct pair of sides.
sin A = 7/25 because sine is opposite over hypotenuse. cos A = 24/25 because cosine is adjacent over hypotenuse. tan A = 7/24 because tangent is opposite over adjacent. - 2
A right triangle has an acute angle of 35 degrees. The hypotenuse is 18 cm. Which trig ratio should you use to find the side opposite the 35 degree angle, and what is the length of that side?
Opposite and hypotenuse means use SOH.
Use sine because the opposite side and hypotenuse are involved. sin 35° = x/18, so x = 18 sin 35° ≈ 10.3 cm. - 3
A right triangle has an acute angle of 62 degrees. The side adjacent to the angle is 9 inches. Find the hypotenuse.
Use cosine because adjacent and hypotenuse are involved. cos 62° = 9/h, so h = 9/cos 62° ≈ 19.2 inches. - 4
A ladder leans against a wall and makes a 70 degree angle with the ground. The ladder is 15 feet long. How high up the wall does the ladder reach?
The ladder is the longest side of the right triangle.
Use sine because the height is opposite the 70 degree angle and the ladder is the hypotenuse. sin 70° = h/15, so h = 15 sin 70° ≈ 14.1 feet. - 5
A ramp rises 4 feet over a horizontal distance of 12 feet. What is the angle of elevation of the ramp to the nearest degree?
Angle of elevation is measured from the horizontal ground.
Use tangent because the rise is opposite the angle and the horizontal distance is adjacent. tan θ = 4/12, so θ = tan⁻¹(4/12) ≈ 18°. - 6
For an acute angle B in a right triangle, sin B = 5/13. If the hypotenuse is 26 units, find the side opposite angle B.
Since sin B = opposite/hypotenuse, the ratio is 5/13. Scaling the hypotenuse from 13 to 26 multiplies by 2, so the opposite side is 10 units. - 7
A right triangle has an acute angle of 48 degrees and an adjacent side of 14 meters. Find the length of the opposite side.
Tangent compares opposite to adjacent.
Use tangent because opposite and adjacent are involved. tan 48° = x/14, so x = 14 tan 48° ≈ 15.5 meters. - 8
A cable is attached from the top of a pole to a point on the ground 20 feet from the base of the pole. The cable makes a 38 degree angle with the ground. How tall is the pole?
Use tangent because the pole height is opposite the 38 degree angle and the ground distance is adjacent. tan 38° = h/20, so h = 20 tan 38° ≈ 15.6 feet. - 9
In a right triangle, angle C is acute. The opposite side is 16 and the adjacent side is 30. Find angle C to the nearest degree.
Use an inverse trig function because you are finding the angle.
Use tangent because opposite and adjacent are known. tan C = 16/30, so C = tan⁻¹(16/30) ≈ 28°. - 10
A right triangle has an acute angle of 25 degrees and a hypotenuse of 40 centimeters. Find the side adjacent to the 25 degree angle.
Adjacent and hypotenuse means use CAH.
Use cosine because adjacent and hypotenuse are involved. cos 25° = x/40, so x = 40 cos 25° ≈ 36.3 centimeters. - 11
A boat is 85 meters from the base of a lighthouse. The angle of elevation from the boat to the top of the lighthouse is 12 degrees. Find the height of the lighthouse.
Use tangent because the lighthouse height is opposite the angle and the horizontal distance is adjacent. tan 12° = h/85, so h = 85 tan 12° ≈ 18.1 meters. - 12
A right triangle has legs of 8 and 15 and a hypotenuse of 17. For the acute angle opposite the side of length 8, write the sine, cosine, and tangent ratios as fractions.
First identify which leg is opposite the given angle. The other leg is adjacent.
For the angle opposite the side of length 8, sin θ = 8/17, cos θ = 15/17, and tan θ = 8/15.