Understanding Sine, Cosine, and Tangent
Sine, cosine, and tangent are trigonometric functions that relate angles to ratios in right triangles. They also have elegant geometric meanings on the unit circle—a circle with radius 1 centred at the origin.
On the unit circle, if you draw a ray from the centre at angle α:
- Sine (sin α) measures the vertical distance from the x-axis (the y-coordinate).
- Cosine (cos α) measures the horizontal distance from the y-axis (the x-coordinate).
- Tangent (tan α) represents the slope of the line, or the ratio sin α ÷ cos α. It is tangent to the unit circle, hence the name.
The reciprocal functions—cosecant (csc), secant (sec), and cotangent (cot)—are inverses of sine, cosine, and tangent respectively. They become undefined when their corresponding primary functions equal zero.
Core Trigonometric Ratios in Right Triangles
In a right triangle with an angle α, label the sides as follows: the side across from α is opposite, the side adjacent to α is adjacent, and the longest side is the hypotenuse. The fundamental trigonometric ratios are:
sin(α) = opposite ÷ hypotenuse
cos(α) = adjacent ÷ hypotenuse
tan(α) = opposite ÷ adjacent
tan(α) = sin(α) ÷ cos(α)
csc(α) = hypotenuse ÷ opposite
sec(α) = hypotenuse ÷ adjacent
cot(α) = adjacent ÷ opposite
opposite— The side of the triangle opposite to angle α.adjacent— The side of the triangle adjacent to (touching) angle α.hypotenuse— The longest side of the right triangle, opposite the right angle.α (alpha)— The angle of interest in the right triangle.
Complementary Angles and Special Identities
In any right triangle, the two non-right angles are complementary—they sum to 90°. This creates a powerful relationship:
- sin(90° − α) = cos(α)
- cos(90° − α) = sin(α)
- tan(90° − α) = cot(α)
These identities mean that if you know one acute angle's sine, you automatically know the other angle's cosine. For instance, in a 3-4-5 triangle with the 3-4 angle: sin(α) = 3/5, so for the complementary angle β, sin(β) = 4/5. Additionally, the Pythagorean identity sin²(α) + cos²(α) = 1 holds for all angles and is fundamental to trigonometry.
Common Pitfalls and Practical Notes
Understanding trigonometric behaviour across different angle ranges prevents calculation errors.
- Tangent and Cotangent Undefined Points — Tangent is undefined at 90°, 270°, and every 180° thereafter because cos(α) becomes zero in the denominator. Cotangent is undefined at 0°, 180°, 360° and every 180° because sin(α) equals zero. Always check your angle before assuming these functions exist.
- Angle Unit Confusion — Trigonometric functions behave differently in degrees versus radians. An angle of 180° equals π radians, not 180 π. Most calculators and programming languages default to radians, so verify your input mode. π/6 radians equals 30°, not 180°.
- Negative Angles and Periodicity — Trigonometric functions repeat every 360° (or 2π radians). Sine and cosine have this period; tangent and cotangent repeat every 180° (or π radians). Negative angles wrap around the circle backwards, so sin(−α) = −sin(α), but cos(−α) = cos(α).
- Precision Near Zero — When an angle approaches values where sine or cosine is nearly zero (like 90° for cosine or 0° for sine), rounding errors can inflate or deflate the result. Use this calculator's built-in cleanup to avoid spurious output like 1e−15 instead of exactly 0.
Solving Right Triangles with Trigonometry
With at least two pieces of information about a right triangle (two sides, or one side and one acute angle), you can find everything else using these functions.
- Given two sides: Use the Pythagorean theorem to find the third, then inverse trigonometric functions (arcsin, arccos, arctan) to find angles.
- Given one side and one acute angle: Use sin, cos, or tan to find a second side. Then use the Pythagorean theorem or complementary angle to find the third.
- Given an area and one side: Calculate the other side from the area formula (Area = ½ × side₁ × side₂), then find the hypotenuse and remaining angle.
In a 3-4-5 triangle, sin(α) = 3/5 = 0.6 for the angle opposite the side of length 3. Because sin²(α) + cos²(α) = 1, we have cos(α) = 4/5 = 0.8. Then tan(α) = 3/4 = 0.75. For the other acute angle (the complement), all values flip: sin swaps with cos, and tan becomes its reciprocal.