Educator

A right triangle has short sides of length 5 and 12. Find all the angles in the triangle

• Start by drawing a right triangle with labeled side lengths and angles
• 
• Recall SOHCAHTOA, sin x = [Opposite/Hypotenuse], cos x = [Adjacent/Hypotenuse], tan x = [Opposite/Adjacent]
• tanθ = [5/12] ⇒ θ = arctan([5/12]) ⇒ θ ≈ 22.6^° (Make sure your calculator is in degree mode)
• tanϕ = [12/5] ⇒ ϕ = arctan([12/5]) ⇒ ϕ ≈ 67.4^°

Check your answers: 90^° + 22.6^° + 67.4^° = 180^°

A right triangle has angle measure 33^° and adjacent length 7. Find the missing angle and the lengths of the other two sides of the triangle

• Start by drawing a right triangle with labeled side lengths and angles
• 
• Recall SOHCAHTOA, sin x = [Opposite/Hypotenuse], cos x = [Adjacent/Hypotenuse], tan x = [Opposite/Adjacent]

Missing angle: θ = 180^° − 90^° − 33^° ⇒ θ = 57^°

sin33^° = [x/7] ⇒ x = 7sin(33^°) ⇒ x ≈ 3.8 (Make sure your calculator is in degree mode

cos33^° = [7/y] ⇒ y = [7/(cos33^°)] ⇒ y ≈ 8.3

The lengths of two short sides of a right triangle are in a 7:4 ratio. Find the missing side length and all the angles of the triangle

• Start by drawing a right triangle with labeled side lengths and angles
• 
• Recall SOHCAHTOA, sin x = [Opposite/Hypotenuse], cos x = [Adjacent/Hypotenuse], tan x = [Opposite/Adjacent]
• Missing side length: Use Pythagorean Theorem a² + b² = c²

Missing side length: x² = 7² + 4² ⇒ x² = 49 + 16 ⇒ x² = 65 ⇒ x = √{65}

tanθ = [7/4] ⇒ θ = arctan([7/4]) ⇒ θ ≈ 60.3^°

sinϕ = [4/(√{65} )] ⇒ ϕ = arcsin([4/(√{65} )]) ⇒ ϕ ≈ 29.7^°

A right triangle has short side of length 4 and hypotenuse of length 9. Find the missing side length and all the angles of the triangle

• Start by drawing a right triangle with labeled side lengths and angles
• 
• Recall SOHCAHTOA, sin x = [Opposite/Hypotenuse], cos x = [Adjacent/Hypotenuse], tan x = [Opposite/Adjacent]
• Missing side length: Use Pythagorean Theorem a² + b² = c²

Missing side length: 9² = x² + 4² ⇒ 81 = x² + 16 ⇒ x² = 65 ⇒ x = √{65}

sinθ = [4/9] ⇒ θ = arcsin([4/9]) ⇒ θ ≈ 26.4^°

cosϕ = [4/9] ⇒ ϕ = arccos([4/9]) ⇒ ϕ ≈ 63.6^°

A right triangle has one angle of 56^° and hypotenuse of length 7. Find the lengths of all the sides and the missing angle

• Start by drawing a right triangle with labeled side lengths and angles
• 
• Recall SOHCAHTOA, sin x = [Opposite/Hypotenuse], cos x = [Adjacent/Hypotenuse], tan x = [Opposite/Adjacent]

Missing angle: θ = 180^° − 90^° − 56^° ⇒ θ = 34^°

sin56^° = [x/7] ⇒ x = 7sin56^° ⇒ x ≈ 5.8

cos56^° = [y/7] ⇒ y = 7cos56^° ⇒ y ≈ 3.9

A right triangle has short sides of length 11 and 12. Find all the angles in the triangle

• Start by drawing a right triangle with labeled side lengths and angles
• 
• Recall SOHCAHTOA, sin x = [Opposite/Hypotenuse], cos x = [Adjacent/Hypotenuse], tan x = [Opposite/Adjacent]
• tanθ = [12/11] ⇒ θ = arctan([12/11]) ⇒ θ ≈ 47.5^° (Make sure your calculator is in degree mode)
• tanϕ = [11/12] ⇒ ϕ = arctan([11/12]) ⇒ ϕ ≈ 42.5^°

Check your answers: 90^° + 42.5^° + 47.5^° = 180^°

A right triangle has angle measure 27^° and opposite side length 4. Find the missing angle and the lengths of the other two sides of the triangle

• Start by drawing a right triangle with labeled side lengths and angles
• 
• Recall SOHCAHTOA, sin x = [Opposite/Hypotenuse], cos x = [Adjacent/Hypotenuse], tan x = [Opposite/Adjacent]

Missing angle: θ = 180^° - 90^° - 27^° ⇒ θ = 57^°

sin27^° = [4/y] ⇒ y = [4/(sin27^°)] ⇒ y ≈ 8.8 (Make sure your calculator is in degree mode)

tan27^° = [4/x] ⇒ x = [4/(tan27^°)] ⇒ x ≈ 7.9

The lengths of two short sides of a right triangle are in a 6:3 ratio. Find the missing side length and all the angles of the triangle

• Start by drawing a right triangle with labeled side lengths and angles
• 
• Recall SOHCAHTOA, sin x = [Opposite/Hypotenuse], cos x = [Adjacent/Hypotenuse], tan x = [Opposite/Adjacent]
• Missing side length: Use Pythagorean Theorem a² + b² = c²

Missing side length: x² = 3² + 6² ⇒ x² = 9 + 36 ⇒ x² = 45 ⇒ x = √{45}

tanθ = [6/3] ⇒ θ = arctan([6/3]) ⇒ θ ≈ 63.4^°

tanϕ = [3/6] ⇒ ϕ = arctan([3/6]) ⇒ ϕ ≈ 26.6^°

A right triangle has short side of length 7 and hypotenuse of length 11. Find the missing side length and all the angles of the triangle

• Start by drawing a right triangle with labeled side lengths and angles
• 
• Recall SOHCAHTOA, sin x = [Opposite/Hypotenuse], cos x = [Adjacent/Hypotenuse], tan x = [Opposite/Adjacent]
• Missing side length: Use Pythagorean Theorem a² + b² = c²

Missing side length: 11² = x² + 7² ⇒ 121 = x² + 49 ⇒ x² = 72 ⇒ x = 6√2

sinθ = [7/11] ⇒ θ = arcsin([7/11]) ⇒ θ ≈ 39.5^°

cosϕ = [7/11] ⇒ ϕ = arccos([7/11]) ⇒ ϕ ≈ 50.5^°

A right triangle has one angle of 68^° and hypotenuse of length 9. Find the lengths of all the sides and the missing angle

• Start by drawing a right triangle with labeled side lengths and angles
• 
• Recall SOHCAHTOA, sin x = [Opposite/Hypotenuse], cos x = [Adjacent/Hypotenuse], tan x = [Opposite/Adjacent]

Missing angle: θ = 180^° − 90^° − 68^° ⇒ θ = 22^°

sin68^° = [x/9] ⇒ x = 9sin68^° ⇒ x ≈ 8.3

cos68^° = [y/9] ⇒ y = 9cos68^° ⇒ y ≈ 3.4

1. Intro
2. Master Formula for Right Angles
• SOHCAHTOA
• Only for Right Triangles
3. Example 1: Find All Angles in a Triangle
4. Example 2: Find Lengths of All Sides of Triangle
5. Example 3: Find All Angles in a Triangle
6. Extra Example 1: Find All Angles in a Triangle
7. Extra Example 2: Find Lengths of All Sides of Triangle