Right Triangle Trigonometry

What Is a Right Triangle?

A right triangle is any triangle that contains exactly one 90-degree angle, often marked with a small square in diagrams. The side directly opposite the right angle is called the hypotenuse — it is always the longest side of the triangle. The other two sides are referred to as legs.

What makes right triangles so important in trigonometry is that the ratios between their sides are entirely determined by the measures of their acute angles. If you know one acute angle and any one side, you can find every other measurement in the triangle. This predictable relationship is the foundation of all trigonometry.

In the diagram below, angle C is the right angle (90°). The side labeled c (opposite the right angle) is the hypotenuse. Relative to angle A, side a is the opposite side and side b is the adjacent side. These labels change depending on which angle you are referencing — a critical detail that many students overlook.

The Three Primary Trig Ratios (SOH-CAH-TOA)

The three fundamental trigonometric ratios — sine, cosine, and tangent — each describe a specific relationship between two sides of a right triangle relative to one of its acute angles. The mnemonic SOH-CAH-TOA is the most widely used tool for remembering them:

Here, θ (theta) represents whichever acute angle you are working with. Let us break down each ratio:

Sine (sin)

The sine of an angle is the ratio of the length of the side opposite that angle to the length of the hypotenuse. In our diagram, sin(A) = a / c. The sine function always produces a value between 0 and 1 for acute angles. As the angle increases from 0° toward 90°, its sine increases from 0 toward 1.

Cosine (cos)

The cosine of an angle is the ratio of the length of the side adjacent to that angle to the length of the hypotenuse. In our diagram, cos(A) = b / c. Like sine, cosine produces values between 0 and 1 for acute angles, but it decreases as the angle increases. Notice that sin(A) = cos(B) and cos(A) = sin(B) — complementary angles always have this relationship.

Tangent (tan)

The tangent of an angle is the ratio of the side opposite that angle to the side adjacent to it. In our diagram, tan(A) = a / b. Unlike sine and cosine, tangent is not bounded between 0 and 1. It can take any positive value and increases without bound as the angle approaches 90°. Tangent can also be expressed as sin(θ) / cos(θ).

Finding Missing Sides

When you know one acute angle and one side of a right triangle, you can find any other side by choosing the appropriate trig ratio. The process follows a consistent pattern:

1. Identify which sides you know and which you need to find.
2. Label the sides as opposite, adjacent, or hypotenuse relative to the known angle.
3. Choose the trig ratio that connects the known side to the unknown side.
4. Set up the equation and solve algebraically.

Finding Missing Angles

When you know two sides of a right triangle but not the acute angles, you use inverse trigonometric functions to work backwards from a ratio to an angle. The three inverse functions are:

On most calculators, these are labeled sin⁻¹, cos⁻¹, and tan⁻¹ (or sometimes arcsin, arccos, and arctan). The process is straightforward: compute the ratio of the two known sides, then apply the appropriate inverse function.

Remember that the two acute angles in a right triangle always sum to 90°. So once you find one acute angle, the other is simply 90° minus that angle. In the example above, the other acute angle would be 90° − 35.0° = 55.0°.

Special Right Triangles

Two families of right triangles appear so frequently in mathematics that their side ratios are worth memorizing. These special right triangles let you solve problems exactly, without a calculator.

The 45-45-90 Triangle

A 45-45-90 triangle is an isosceles right triangle, meaning the two legs are equal in length. If each leg has length 1, the hypotenuse has length √2 (by the Pythagorean theorem: 1² + 1² = 2). The general side ratio is:

For any 45-45-90 triangle with legs of length s, the hypotenuse is s√2. Conversely, if the hypotenuse is h, each leg is h / √2 = h√2 / 2. This triangle tells us that sin(45°) = cos(45°) = √2 / 2 ≈ 0.7071, and tan(45°) = 1.

The 30-60-90 Triangle

A 30-60-90 triangle is formed by cutting an equilateral triangle in half. The side ratio is:

The shortest side (opposite the 30° angle) has length 1, the longer leg (opposite the 60° angle) has length √3, and the hypotenuse (opposite the 90° angle) has length 2. This gives us several exact trig values:

• sin(30°) = 1/2,   cos(30°) = √3/2,   tan(30°) = 1/√3 = √3/3
• sin(60°) = √3/2,   cos(60°) = 1/2,   tan(60°) = √3

Real-World Applications

Right triangle trigonometry is not an abstract exercise. It is used daily in a wide range of professions and everyday situations:

• Architecture and construction: Builders calculate roof pitch (the angle of a roof slope) using tangent. A roof that rises 6 inches for every 12 inches of horizontal run has a pitch angle of tan¹(6/12) = 26.6°.
• Navigation and surveying: Surveyors measure distances across rivers, valleys, and other obstacles by sighting angles and measuring one accessible baseline, then computing the unknown distances with trig ratios.
• Aviation: Pilots use angles of descent and climb. If an aircraft descends at a 3° glide slope and needs to lose 5,000 feet of altitude, the ground distance covered is 5000 / tan(3°) ≈ 95,441 feet, or about 15.6 nautical miles.
• Physics: Resolving forces into horizontal and vertical components requires sine and cosine. A 100 N force applied at 40° above the horizontal has a horizontal component of 100 × cos(40°) ≈ 76.6 N and a vertical component of 100 × sin(40°) ≈ 64.3 N.
• Everyday life: Determining the height of a tree, a building, or a flagpole by measuring the angle of elevation from a known distance away is a direct application of the tangent ratio.

Common Mistakes to Avoid

Even experienced students fall into these traps. Being aware of them will save you marks on tests and prevent errors in practice:

• Mixing up opposite and adjacent: These labels change depending on which angle you are working with. Always re-label the sides for the specific angle in the problem before choosing a trig ratio.
• Using trig ratios on non-right triangles: SOH-CAH-TOA only works for right triangles. For other triangles, you need the Law of Sines or Law of Cosines.
• Calculator in the wrong mode: If your problem uses degrees but your calculator is set to radians (or vice versa), every answer will be incorrect. Double-check the mode before computing.
• Forgetting to use the inverse function: When finding an angle, you need sin⁻¹, cos⁻¹, or tan⁻¹. Simply computing sin(0.7) does not give you the angle whose sine is 0.7. You need sin⁻¹(0.7).
• Rounding too early: Keep at least four decimal places in intermediate calculations and only round your final answer. Early rounding compounds errors and leads to inaccurate results.
• Confusing the hypotenuse with a leg: The hypotenuse is always opposite the 90° angle and is always the longest side. If your calculated side comes out longer than the hypotenuse, you have made an error.

Practice Problems