Sine, Cosine, and Tangent

Objective: Identify the correct side ratio for an acute right-triangle angle.

Prerequisite skills: right-triangle vocabulary and ratios. Sine is opposite over hypotenuse; cosine is adjacent over hypotenuse. Changing the reference acute angle exchanges the opposite and adjacent legs but keeps the same hypotenuse. Common error: exchanging sine and cosine or using a leg as the denominator instead of the hypotenuse. Restrictions and interpretation: Define opposite and adjacent relative to the selected acute angle; the hypotenuse is always opposite 90°.

Objective: Use the ratio of the two legs.

Tangent is opposite divided by adjacent. Because sine and cosine share the hypotenuse denominator, tanθ=sinθ/cosθ when cosine is nonzero. Acute right-triangle angles have positive side ratios. Common error: inverting the two legs for the chosen reference angle. Restrictions and interpretation: Define opposite and adjacent relative to the selected acute angle; the hypotenuse is always opposite 90°.

Objective: Select sine, cosine, or tangent using the known and requested sides.

Name which two side roles are involved before writing a ratio. If the hypotenuse and a sine are known, multiply the hypotenuse by sine to obtain the opposite leg. If solving for the hypotenuse, divide instead. Common error: dividing when the unknown is a numerator leg. Restrictions and interpretation: Define opposite and adjacent relative to the selected acute angle; the hypotenuse is always opposite 90°. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.