Trigonometric ratios: find the length of a side

Learning Objectives Identify the hypotenuse, opposite, and adjacent sides of a right-angled triangle with respect to a given acute angle. Recall and state the three primary trigonometric ratios: sine, cosine, and tangent (SOH CAH TOA). Select the appropriate trigonometric ratio to use based on the given side and angle information. Set up a correct trigonometric equation to model a problem. Algebraically solve the trigonometric equation for an unknown side length. Use a scientific calculator to correctly evaluate trigonometric functions and find a final answer. How can you find the height of a giant tree or a skyscraper without actually climbing it? 🌳 In this tutorial, you will learn how to use the powerful tools of trigonometry—sine, cosine, and tangent—to find the missing...

TermDefinitionExample Right-Angled TriangleA triangle that has one angle measuring exactly 90 degrees.A triangle with angles 30°, 60°, and 90°. Reference Angle (θ)The specific acute angle (less than 90°) in a right-angled triangle that we are focusing on for our calculations.In a problem stating 'from a 40° angle of elevation', the reference angle θ is 40°. HypotenuseThe longest side of a right-angled triangle. It is always the side directly opposite the 90-degree angle.In a triangle with sides 3, 4, and 5, the side with length 5 is the hypotenuse. Opposite SideThe side directly across from the reference angle (θ). Its identity depends on which acute angle you choose.If your reference angle is 30°, the side across from that angle is the 'opposite' side. Adjacent SideTh...

The Sine Ratio (SOH) \sin(\theta) = \frac{\text{Opposite}}{\text{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(\theta) = \frac{\text{Adjacent}}{\text{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(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} Use this ratio when you know or need to find the Opposite and Adjacent sides, relative to the reference angle θ.