The Tangent Ratio
Part of the Right-Triangle Trigonometry section of Coddy's Trigonometry journey. Lesson 9 of 50.
The tangent of θ leaves the hypotenuse out: it is the opposite side divided by the adjacent side.
tan θ = opposite / adjacent
In the 3, 4, 5 triangle with θ opposite the 3, tan(theta) = 3/4. For the other acute angle the legs swap, so its tangent is 4/3, which is more than 1. When the two legs are equal, both acute angles are 45° and tan(45°) = 1. Sine and cosine divide by the longest side and stay below 1; tangent divides one leg by the other and can be any positive number.
Tangent is the slope you already know. A road that climbs 3 m for every 4 m along the flat rises at the angle whose tangent is 3/4: rise over run.
Challenge
Triangle DEF has a right angle at D and a 45° angle at E. The side next to the 45° angle is 11. Find the side x opposite the 45° angle: tap the angle and both sides, pick the ratio, and solve. Then press Check.
Try it yourself
Use the figure to find x
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Right-Triangle Trigonometry
1Sides of a Right Triangle
How Tall Is the Tree?The HypotenuseThe Opposite SideThe Adjacent SideWhen the Angle MovesRecap: Naming the Sides2The Three Ratios
The Sine RatioThe Cosine RatioThe Tangent RatioChoosing the RatioRecap: The Three Ratios