Trigonometric ratios find a side length

Learning Objectives Identify the hypotenuse, opposite, and adjacent sides of a right-angled triangle relative to a given reference angle. Select the appropriate trigonometric ratio (sine, cosine, or tangent) to solve for an unknown side length. Construct a correct trigonometric equation using the given angle and side length. Algebraically manipulate the trigonometric equation to isolate the variable representing the unknown side. Calculate the unknown side length using a calculator, ensuring it is in the correct angular mode (degrees or radians). Apply trigonometric ratios to solve for side lengths in applied problems, such as those involving angles of elevation and depression. How can a surveyor measure the width of a river without ever crossing it? 🏞️ The answer lies in th...

TermDefinitionExample Reference Angle (θ)The acute angle in a right-angled triangle that is used as the point of reference for identifying the opposite and adjacent sides.In a triangle with angles 30°, 60°, and 90°, if you are working with the 30° angle, then θ = 30°. Opposite SideThe side of the triangle that is directly across from the reference angle (θ).If your reference angle is at the bottom-left vertex, the opposite side is the vertical side on the right. Adjacent SideThe side of the triangle that is next to the reference angle (θ), but is not the hypotenuse.If your reference angle is at the bottom-left vertex, the adjacent side is the horizontal side at the bottom. HypotenuseThe longest side of a right-angled triangle, located directly opposite the 90° angle.In any right-angled tr...

The Sine Ratio (SOH) sin(θ) = Opposite / Hypotenuse Use this ratio when you know or need to find the opposite side and the hypotenuse, relative to the reference angle θ. The Cosine Ratio (CAH) cos(θ) = Adjacent / Hypotenuse Use this ratio when you know or need to find the adjacent side and the hypotenuse, relative to the reference angle θ. The Tangent Ratio (TOA) tan(θ) = Opposite / Adjacent Use this ratio when you know or need to find the opposite and adjacent sides, relative to the reference angle θ.