How to solve for x in a right triangle

How to Solve a Right Triangle

Step 1: Determine which sides (adjacent, opposite, or hypotenuse) are known in relation to the given angle.

Step 2: Set up the proper equation with the trigonometric rule that relates the known side to the unknown side.

Step 3: Solve the equation for the unknown side.

How to Solve a Right Triangle: Vocabulary

Right Triangle: A right triangle is a triangle that contains a 90-degree angle. The other two angles must be acute and their sum is equal to 90 degrees. In a right triangle, the side opposite the right angle is known as the hypotenuse. The other two legs of the triangle are also named for their relationship to one of the two acute angles. The side that forms the identified acute angle with the hypotenuse is known as the adjacent side and the side opposite the identified angle is known as the opposite side. The diagram below illustrates these sides with their names:

Trigonometric Functions: There are three trigonometric functions that define the relationship between two of the three sides of a triangle: sine (sin), cosine (cos), and tangent (tan). Each function takes the marked angle as its input and returns the ratio of two of the sides:

$$\sin{\theta} = \dfrac{\rm{opposite}}{\rm{hypotenuse}} \\\\ \cos{\theta} = \dfrac{\rm{adjacent}}{\rm{hypotenuse}} \\\\ \tan{\theta} = \dfrac{\rm{opposite}}{\rm{adjacent}} $$

The following two problems demonstrate how to solve a right triangle.

How to Solve a Right Triangle: Example 1

Solve the following triangle for {eq}x {/eq}:

Step 1: We must first identify the known and unknown sides. The known side of length 12 is adjacent to the marked angle. The unknown side, {eq}x {/eq} is the hypotenuse because it is opposite the right angle.

Step 2: The trigonometric function that relates the adjacent side to the hypotenuse is cosine. Therefore, we can set up an equation using the cosine function:

$$\cos{20 ^\circ} = \dfrac{12}{x} $$

Step 3: To solve for {eq}x {/eq}, we will first multiply both sides by {eq}x {/eq} to get it out of the denominator:

$$x \cos{20 ^\circ} = \dfrac{12}{x} \times x $$

The {eq}x {/eq} on the right side cancels, so we have:

$$x \cos{20 ^\circ} = 12 $$

We will now divide both sides by {eq}\cos{20 ^\circ} {/eq}:

$$\dfrac{x \cos{20 ^\circ}}{\cos{20 ^\circ}} = \dfrac{12}{\cos{20 ^\circ}} $$

The cosine term cancels from the left side, so our solution of this equation is:

$$x = \dfrac{12}{\cos{20 ^\circ}} \approx 12.8 $$

The hypotenuse has a length of 12.8 units.

How to Solve a Right Triangle: Example 2

Solve the following right triangle for {eq}x {/eq}:

Step 1: The adjacent side is the unknown side. The opposite side has a length of 4 units.

Step 2: The trigonometric function that relates the opposite and adjacent sides is tangent. Therefore, the equation is:

$$\tan{50 ^\circ} = \dfrac{4}{x} $$

Step 3: Solving this equation for {eq}x {/eq}, we will first multiply both sides by {eq}x {/eq}:

$$x \tan{50 ^\circ} = 4 $$

Then we will divide by {eq}\tan{50 ^\circ} {/eq}:

$$x = \dfrac{4}{\tan{50 ^\circ}} \approx 3.4 $$

The unknown side has a length of 3.4 units.

Get access to thousands of practice questions and explanations!

There are many ways to find the side length of a right triangle. We are going to focus on two specific cases.

Case II

We know 1 side and 1 angle of the right triangle, in which case, use sohcahtoa.

Video Tutorial on Finding the Side Length of a Right Triangle

Practice Problems

Calculate the length of the sides below. In each case, round your answer to the nearest hundredth.

Problem 1

Find the length of side X in the triangle below.

Step 2

Substitute the two known sides into the Pythagorean theorem's formula:

$$ a^2 + b^2 = c^2 \\ 8^2 + 6^2 = x^2 \\ 100 = x^2 \\ x = \sqrt{100} \\ x = \boxed{10} $$

Problem 2

Find the length of side X in the right triangle below.

Step 1

Since we know 1 side and 1 angle of this triangle, we will use sohcahtoa.

Step 2

Set up an equation using a sohcahtoa ratio. Since we know the hypotenuse and want to find the side opposite of the 53° angle, we are dealing with sine

$$ sin(53) = \frac{ opposite}{hypotenuse} \\ sin(53) = \frac{ \red x }{ 12 } $$

Now, just solve the Equation:

Step 3

$$ sin(53) = \frac{ \red x }{ 12 } \\ \red x = 12 \cdot sin (53) \\ \red x = \boxed{ 11.98} $$

Problem 3

Find the length of side X in the right triangle below.

Step 2

Substitute the two known sides into the Pythagorean theorem's formula:

$$ a^2 + b^2 = c^2 \\ \red t^2 + 12^2 = 13^2 \\ \red t^2 + 144 = 169 \\ \red t^2 = 169 - 144 \\ \red t^2 = 25 \\ \red t = \boxed{5} $$

Problem 4

Find the length of side X in the right triangle below.

Step 1

Since we know 1 side and 1 angle of this triangle, we will use sohcahtoa.

Step 2

Set up an equation using the sine, cosine or tangent ratio Since we want to know the length of the hypotenuse, and we already know the side opposite of the 53° angle, we are dealing with sine.

$$ sin(67) = \frac{opp}{hyp} \\ sin(67) = \frac{24}{\red x} $$

Now, just solve the Equation:

Step 3

$$ x = \frac{ 24}{ sin(67) } \\ x = 26.07 $$

Problem 5

Calculate the length of side X in the right triangle below.

Step 1

Since we know 2 sides and 1 angle of this triangle, we can use either the Pythagorean theorem (by making use of the two sides) or use sohcahtoa (by making use of the angle and 1 of the given sides).

Step 2

Chose which way you want to solve this problem. There are several different solutions. The only thing you cannot use is sine, since the sine ratio does not involve the adjacent side, x, which we are trying to find.

The answers are slightly different (tangent s 35.34 vs 36 for the others) due to rounding issues. I rounded the angle's measure to 23° for the sake of simplicity of the diagram. A more accurate angle measure would have been 22.61986495°. If you use that value instead of 23°, you will get answers that are more consistent.

Step 3

$$ x = \frac{ 24}{ sin(67) } \approx 26.07 $$