Sine, cosine and tangent
The three ratios that connect angles with sides
What is this about?
In a right triangle, trigonometric ratios relate an angle to the quotient between two sides. Looking from the angle: the opposite side is the one across from it, the adjacent side touches it, and the hypotenuse is always the longest side. Sine, cosine and tangent are just divisions between those three sides.
The rule to remember
SOH-CAH-TOA: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent. First identify the sides FROM the given angle, then apply the right division.
Example 1
sin α with opposite = 3 and hypotenuse = 5
- 1Identify what you're given: the opposite side (3) and the hypotenuse (5).
- 2The ratio that uses those two sides is the sine: sin = opposite ÷ hypotenuse.
- 3Substitute: sin α = 3 ÷ 5 = 0.6.
- 4Quick check: the sine is always between 0 and 1, and 0.6 passes. ✓
Result: 0.6
Example 2
tan β with opposite = 6 and adjacent = 8
- 1You're given both legs: opposite (6) and adjacent (8).
- 2The ratio that relates them is the tangent: tan = opposite ÷ adjacent.
- 3Substitute: tan β = 6 ÷ 8 = 0.75.
- 4The tangent CAN exceed 1 (when the opposite is longer than the adjacent); here it doesn't.
Result: 0.75
The most common mistake
Many people write
sin α = hypotenuse ÷ opposite = 5/3
Why it's wrong
The division is flipped: sine is opposite OVER hypotenuse, not the other way around. Since the hypotenuse is the longest side, the sine can never exceed 1, and 5/3 ≈ 1.67 warns you something is wrong. The correct value is 3/5 = 0.6.
Did it click?
The best way to find out is to try. These exercises are exactly on this topic.
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