Find Angles from Sides with arcsin, arccos, arctan in Applications

1) A right triangle has opposite side 3 and adjacent side 4. What is the acute angle?

Arcsin(4/5)

Arctan(4/3)

Arctan(3/4)

Arccos(3/5)

Use tanθ = opposite/adjacent = 3/4 ⇒ θ = arctan(3/4).

Explanation

Use tanθ = opposite/adjacent = 3/4 ⇒ θ = arctan(3/4).

2) A rectangular screen is 12 cm wide and 5 cm tall. What is the angle from the base to the top corner?

Arctan(12/5)

Arctan(5/12)

Arcsin(12/13)

Arccos(5/13)

Angle above base: tanθ = 5/12 ⇒ θ = arctan(5/12).

Explanation

Angle above base: tanθ = 5/12 ⇒ θ = arctan(5/12).

3) A ladder is 8 m long and touches the wall 6 m high. What is the angle with the ground?

Arccos(6/8)

Arccos(8/6)

Arcsin(6/10)

Arctan(6/√28)

Opposite = 6, adjacent = √(8²−6²)=√28 ⇒ tanθ = 6/√28 ⇒ θ = arctan(6/√28).

Explanation

Opposite = 6, adjacent = √(8²−6²)=√28 ⇒ tanθ = 6/√28 ⇒ θ = arctan(6/√28).

4) A right triangle has adjacent side 7 and hypotenuse 25. What is the acute angle?

Arccos(7/25)

Arccos(25/7)

Arcsin(7/25)

Arctan(7/24)

cosθ = adjacent/hypotenuse = 7/25 ⇒ θ = arccos(7/25).

Explanation

cosθ = adjacent/hypotenuse = 7/25 ⇒ θ = arccos(7/25).

5) A right triangle has adjacent side 7 and hypotenuse 25. What is the acute angle?

Arcsin(7/25)

Arccos(25/7)

Arccos(7/25)

Arctan(7/24)

cosθ = 7/25 ⇒ θ = arccos(7/25).

Explanation

cosθ = 7/25 ⇒ θ = arccos(7/25).

6) A rectangle is 9 m long and 12 m tall. What is the angle of its diagonal with the base?

Arctan(12/9)

Arctan(9/12)

Arcsin(9/15)

Arccos(12/15)

Rise/run = 12/9 ⇒ θ = arctan(12/9).

Explanation

Rise/run = 12/9 ⇒ θ = arctan(12/9).

7) A right triangle has opposite side 8 and hypotenuse 17. What is the acute angle?

Arctan(1/2)

Arccos(8/17)

Arcsin(8/17)

Arctan(15/8)

sinθ = 8/17 ⇒ θ = arcsin(8/17).

Explanation

sinθ = 8/17 ⇒ θ = arcsin(8/17).

8) A photo measures 4 cm by 3 cm. What angle does the diagonal make with the base?

Arctan(3/4)

Arctan(4/3)

Arcsin(4/5)

Arccos(3/5)

tanθ = opposite/adjacent = 3/4 ⇒ θ = arctan(3/4).

Explanation

tanθ = opposite/adjacent = 3/4 ⇒ θ = arctan(3/4).

9) A 12 m string is stretched between the ground and a point 9 m above. What is the angle at the ground?

Arctan(√63/9)

Arccos(9/12)

Arctan(9/√63)

Arcsin(√63/12)

Adjacent = √(12²−9²)=√63, tanθ = 9/√63 ⇒ θ = arctan(9/√63).

Explanation

Adjacent = √(12²−9²)=√63, tanθ = 9/√63 ⇒ θ = arctan(9/√63).

10) A right triangle has sides 5, 12, 13. Find the angle opposite side 5.

Arcsin(5/13)

Arccos(5/13)

Arctan(12/5)

sinθ = 5/13 ⇒ θ = arcsin(5/13).

Explanation

sinθ = 5/13 ⇒ θ = arcsin(5/13).

11) A rectangular field measures 20 m by 21 m. What is the angle the diagonal makes with the longer side?

Arcsin(21/29)

Arctan(21/20)

Arctan(20/21)

Arccos(20/29)

Measure from longer side (21): tanθ = opposite/adjacent = 20/21 ⇒ θ = arctan(20/21).

Explanation

Measure from longer side (21): tanθ = opposite/adjacent = 20/21 ⇒ θ = arctan(20/21).

12) A right triangle has opposite 15 and adjacent 20. What is the acute angle?

Arccos(15/25)

Arctan(20/15)

Arcsin(20/25)

Arctan(15/20)

tanθ = 15/20 = 3/4 ⇒ θ = arctan(15/20).

Explanation

tanθ = 15/20 = 3/4 ⇒ θ = arctan(15/20).

13) A 5 m board is placed so the vertical height is 4 m. What is the angle with the base?

Arcsin(3/5)

Arccos(4/5)

Arctan(4/3)

Arctan(3/4)

Opposite = 4, adjacent = 3 ⇒ tanθ = 4/3 ⇒ θ = arctan(4/3).

Explanation

Opposite = 4, adjacent = 3 ⇒ tanθ = 4/3 ⇒ θ = arctan(4/3).

14) A square has side 10 cm. What is the angle the diagonal makes with the side?

Arctan(1)

Arcsin(1/2)

Arccos(1/2)

Arctan(2)

Diagonals of a square are at 45° ⇒ θ = arctan(1).

Explanation

Diagonals of a square are at 45° ⇒ θ = arctan(1).

15) A right triangle has adjacent 9 and hypotenuse 41. What is the acute angle?

Arccos(9/41)

Arccos(41/9)

Arcsin(9/41)

Arctan(9/40)

cosθ = 9/41 ⇒ θ = arccos(9/41).

Explanation

cosθ = 9/41 ⇒ θ = arccos(9/41).

16) A right triangle has opposite 7 and hypotenuse 25. What is the acute angle?

Arcsin(7/25)

Arccos(7/25)

Arctan(7/25)

Arctan(24/7)

sinθ = 7/25 ⇒ θ = arcsin(7/25).

Explanation

sinθ = 7/25 ⇒ θ = arcsin(7/25).

17) A screen measures 16 cm by 12 cm. What is the angle its diagonal makes with the base?

Arcsin(16/20)

Arctan(16/12)

Arctan(12/16)

Arccos(12/20)

Rise/run = 12/16 = 3/4 ⇒ θ = arctan(12/16).

Explanation

Rise/run = 12/16 = 3/4 ⇒ θ = arctan(12/16).

18) A triangle has sides 9, 40, 41. Find the angle opposite side 9.

Arccos(9/41)

Arcsin(40/41)

Arctan(9/40)

Arctan(40/9)

In 9-40-41, tanθ (opposite 9) = 9/40 ⇒ θ = arctan(9/40).

Explanation

In 9-40-41, tanθ (opposite 9) = 9/40 ⇒ θ = arctan(9/40).

19) A right triangle has opposite side 20 and adjacent 21. Find the acute angle.

Arctan(20/21)

Arctan(21/20)

Arcsin(21/29)

Arccos(20/29)

tanθ = 20/21 ⇒ θ = arctan(20/21).

Explanation

tanθ = 20/21 ⇒ θ = arctan(20/21).

20) A rectangle measures 15 m by 8 m. What is the angle the diagonal makes with the longer side?

Arcsin(15/17)

Arctan(15/8)

Arctan(8/15)

Arccos(8/17)

From the longer side (15), rise/run = 8/15 ⇒ θ = arctan(8/15).

Explanation

From the longer side (15), rise/run = 8/15 ⇒ θ = arctan(8/15).

Inverse Trig in Real Life