Right-triangle trigonometry: identify the reference angle

The reference angle determines the side names

In a right triangle, the hypotenuse is opposite the right angle and is the longest side. The other two side names depend on which acute angle you choose. The opposite side lies across from that angle; the adjacent leg touches it but is not the hypotenuse. Sine is opposite divided by hypotenuse, cosine is adjacent divided by hypotenuse, and tangent is opposite divided by adjacent. A ratio has no length unit because the numerator and denominator use the same unit. Identify the sides before choosing a formula, especially when the triangle is described in words.

Choose a ratio from the known and requested quantities

You rarely need all three ratios. If a problem gives a horizontal distance and asks for a vertical height using an elevation angle, tangent often connects the two directly. If it gives a sloping length and asks for a vertical component, sine may be more convenient. The triangle must actually be right-angled for these side-ratio definitions to apply in this form. A calculator's degree or radian setting also matters: a stated angle of 60° is not an input of 60 radians. Exact special-angle ratios can avoid unnecessary rounding.

Worked example 1: the same triangle, different angles

A right triangle has perpendicular legs of 5 cm and 12 cm. Let angle A be opposite the 5 cm leg. Find sine A, cosine A and tangent A. Then consider the other acute angle B. The triangle's physical lengths do not change when the reference angle changes, but which leg is called opposite does change. This is a useful check on whether the side labels were attached to an angle or memorized from a drawing.

• Find the hypotenuse: √(5² + 12²) = √169 = 13 cm.
• Relative to A, opposite = 5 and adjacent = 12. Therefore sin A = 5/13, cos A = 12/13 and tan A = 5/12.
• Relative to B, opposite = 12 and adjacent = 5. Therefore sin B = 12/13, cos B = 5/13 and tan B = 12/5.
• The acute angles are complementary. Swapping the reference angle swaps the sine and cosine ratios, while the hypotenuse stays 13 cm.

Worked example 2: combine two elevation observations

In an ideal geometric model, two observation points lie on level ground along the same straight line from a vertical marker. The farther point is 8 m behind the nearer point. Elevation to the marker's top is 60° from the nearer point and 30° from the farther point. Both observations are made at ground level. Find the marker's height. The common height links the two right triangles; their horizontal distances differ by 8 m.

• Let the nearer horizontal distance be x and the height be h. Then h = x tan 60° = x√3.
• The farther distance is x + 8. Its observation gives h = (x + 8) tan 30° = (x + 8)/√3.
• Equate the heights: x√3 = (x + 8)/√3. Multiplying by √3 gives 3x = x + 8, so x = 4.
• The height is h = 4√3 m, approximately 6.93 m. The farther distance is 12 m, and 12/√3 gives the same height.

Check the geometry before trusting the arithmetic

In the second example, the nearer observer should see the larger elevation angle. That qualitative check agrees with 60° versus 30°. If a calculated distance were negative, you would need to revisit the setup rather than simply remove its sign. If the observation height were above ground, the triangle would measure height above the observer's eye, so that height would need to be added to recover the total. The stated ground-level assumption deliberately removes that extra quantity. Text descriptions can contain enough geometry without requiring a visual diagram.

Misconception: adjacent means any touching side

The hypotenuse touches an acute angle, but it is not the adjacent leg used in tangent. Another common error is using a sloping distance as a horizontal run. Write the relationship in words—vertical over horizontal, for example—before substituting lengths. Keep exact forms such as √3 until the final step when a decimal is requested. Rounding a special-angle ratio early can introduce avoidable error into a multi-step calculation.

Sources & further reading