1 answerLast reply by: Dr. William MurrayMon May 6, 2013 8:54 PMPost by Emily Engle on May 5 at 01:20:29 PMHow would you solve for the inverse ratios without a calculator?

### Trigonometry in Right Angles

Main formulas:

Master formula for right triangles: SOHCAHTOA!

 sinθ = opposite side hypotenuse cosθ = adjacent side hypotenuse tanθ = opposite side adjacent side

Example 1:

A right triangle has short sides of length 3 and 4. Find all the angles in the triangle.

Example 2:

A right triangle has one angle measuring 40°  and opposite side of length 6. Find the lengths of all the sides.

Example 3:

The lengths of the two short sides of a right triangle are in a 5:2 ratio. Find all angles of the triangle.

Example 4:

A right triangle has short sides of length 3 and hypotenuse of length 7. Find all the angles in the triangle.

Example 5:

A right triangle has one angle of 65°  and hypotenuse of length 3. Find the lengths of all the sides of the triangle.

## 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°

## 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]

## 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

## 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

## 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]

## 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°

## 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]

## 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

## 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

## 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

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.