Right Triangle Trigonometry | Mathematics

February 16, 2016 Jennifer Brandt

Trigonometry – a Greek word stemming from trigōnon, "triangle," and metron, "measure," is studied in geometry and involves the relationships between lengths and angles of triangles. In preparation for the TASC Test Assessing Secondary Completion™ Mathematics subtest, we will learn how to solve a right triangle.

What does solving a triangle mean exactly? According to The Math Page, solving for a right triangle means to find the measures of all three sides and all three angles.

The Math Page points out that if ratios of an isosceles right triangle or a 30°-60°-90° triangle (described below) are not known, the three-place trigonometric table is used to solve the triangle.

  • The three-place trigonometric table is a reference that shows you sine, cosine, and tangent values for various angles. We will discuss these terms and use this table in our example below.

Before we solve for a triangle, let’s review some useful information:

  • An isosceles right triangle has two congruent sides and one right angle. The two congruent sides must be the legs (shown below marked with small lines), according to the CK-12 Foundation.

  • A 30°-60°-90° triangle is just as it sounds, a triangle with three separate angles that measure 30 degrees, 60 degrees, and 90 degrees, according to FreeMathHelp.com.

  • The three angles in a right triangle must add up to 180°.
  • When solving for a side of a triangle, you will use formulas called sine, cosine, and tangent. Each one is used differently, depending on the side you’re solving for.
    • The Adjacent side is adjacent to the angle
    • The Opposite is opposite the angle 
    • The longest side is the Hypotenuse

Image via Math Is Fun.

  • Sine: sin(θ) = Opposite / Hypotenuse
  • Cosine: cos(θ) = Adjacent / Hypotenuse
  • Tangent: tan(θ) = Opposite / Adjacent

Right Triangle Trigonometry Example 

Follow The Math Page’s example to learn how to solve for a right triangle:

Example: The following triangle is given one angle (A=36°) and one side (c=10 cm).

Find angle B and sides a and b.

Answer: In every right triangle, the angle with the “box” is the 90° angle. So, since angle A is 36° and angle C is 90°, we can find angle B by subtracting: 90° − 36° = 54° = angle B. To double check your work, see if all three angles add up to 180°.

To find the unknown sides, follow the steps below. Let’s first solve for side a.

  1. Build a fraction with the unknown and known sides, placing the unknown side as the numerator (top of the fraction) and the known side as the denominator (bottom of the fraction).

Unknown
 Known

 = 

 a 
10

  1. Figure out where the side is in relation to the known angle. Is it adjacent, opposite, or the hypotenuse? Since we’re solving for side a, this is opposite of the angle.

Unknown
 Known

 = 

 a 
10

 = 

 Opposite 
Hypotenuse

This side requires the use of sine.

  1. Solve for side a:

 

 

 

  

 a 
10

 = sin 36°

Multiply each side by 10 (the denominator).

(10)

 x 

 a 
10

 = sin 36° x (10)

(a)

= sin 36° x (10)

  1. Use the trigonometric table to determine the sin 36°. Did you get .588?
  2. Solve for side a:

(a)

= .588 x (10)

(a)

= 5.88cm

Can you solve for side b? Work through the same steps and see if you can arrive at the length 8.09 cm. Special note: you will be using the cosine of 36°.

For more practice, watch this video to solve more right triangles. 

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