Grade 10 · Geometry & measurement

Right triangle trig from two ratios practice

Right triangle trig from two ratios is a grade 10 math skill aligned to Common Core standard HSG.SRT.C.6: understand that side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Below are 8 practice questions with answers and step-by-step explanations, drawn from the 10 right triangle trig from two ratios problems our math games drill.

CCSS HSG.SRT.C.610 questions in the bank
Sample questions

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Question 1easy

In a right triangle, sin⁡α=513\sin\alpha = \dfrac{5}{13} and tan⁡α=512\tan\alpha = \dfrac{5}{12}. What is cos⁡α\cos\alpha?

cos⁡α=sin⁡αtan⁡α=513512=513⋅125=1213\cos\alpha = \dfrac{\sin\alpha}{\tan\alpha} = \dfrac{\dfrac{5}{13}}{\dfrac{5}{12}} = \dfrac{5}{13} \cdot \dfrac{12}{5} = \dfrac{12}{13}, or from the 55-1212-1313 triangle: adjacent 1212, hypotenuse 1313.

Question 2easy

In a right triangle, sin⁡α=725\sin\alpha = \dfrac{7}{25} and cos⁡α=2425\cos\alpha = \dfrac{24}{25}. What is tan⁡α\tan\alpha?

tan⁡α=sin⁡αcos⁡α=7252425=725⋅2524=724\tan\alpha = \dfrac{\sin\alpha}{\cos\alpha} = \dfrac{\dfrac{7}{25}}{\dfrac{24}{25}} = \dfrac{7}{25} \cdot \dfrac{25}{24} = \dfrac{7}{24}.

Question 3easy

In a right triangle, sin⁡θ=1213\sin\theta = \dfrac{12}{13} and tan⁡θ=125\tan\theta = \dfrac{12}{5}. What is cos⁡θ\cos\theta?

cos⁡θ=sin⁡θtan⁡θ=1213125=1213⋅512=513\cos\theta = \dfrac{\sin\theta}{\tan\theta} = \dfrac{\dfrac{12}{13}}{\dfrac{12}{5}} = \dfrac{12}{13} \cdot \dfrac{5}{12} = \dfrac{5}{13}.

Question 4easy

In a right triangle, cos⁡α=513\cos\alpha = \dfrac{5}{13} and tan⁡α=125\tan\alpha = \dfrac{12}{5}. What is sin⁡α\sin\alpha?

sin⁡α=tan⁡α⋅cos⁡α=125⋅513=1213\sin\alpha = \tan\alpha \cdot \cos\alpha = \dfrac{12}{5} \cdot \dfrac{5}{13} = \dfrac{12}{13}.

Question 5easy

In a right triangle, sin⁡θ=1517\sin\theta = \dfrac{15}{17} and cos⁡θ=817\cos\theta = \dfrac{8}{17}. What is tan⁡θ\tan\theta?

tan⁡θ=sin⁡θcos⁡θ=1517817=1517⋅178=158\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{\dfrac{15}{17}}{\dfrac{8}{17}} = \dfrac{15}{17} \cdot \dfrac{17}{8} = \dfrac{15}{8}.

Question 6easy

In a right triangle, sin⁡α=45\sin\alpha = \dfrac{4}{5} and tan⁡α=43\tan\alpha = \dfrac{4}{3}. What is cos⁡α\cos\alpha?

cos⁡α=sin⁡αtan⁡α=4543=45⋅34=35\cos\alpha = \dfrac{\sin\alpha}{\tan\alpha} = \dfrac{\dfrac{4}{5}}{\dfrac{4}{3}} = \dfrac{4}{5} \cdot \dfrac{3}{4} = \dfrac{3}{5}.

Question 7easy

In a right triangle, cos⁡θ=1213\cos\theta = \dfrac{12}{13} and tan⁡θ=512\tan\theta = \dfrac{5}{12}. What is sin⁡θ\sin\theta?

sin⁡θ=tan⁡θ⋅cos⁡θ=512⋅1213=513\sin\theta = \tan\theta \cdot \cos\theta = \dfrac{5}{12} \cdot \dfrac{12}{13} = \dfrac{5}{13}.

Question 8easy

In a right triangle, cos⁡α=2425\cos\alpha = \dfrac{24}{25} and tan⁡α=724\tan\alpha = \dfrac{7}{24}. What is sin⁡α\sin\alpha?

sin⁡α=tan⁡α⋅cos⁡α=724⋅2425=725\sin\alpha = \tan\alpha \cdot \cos\alpha = \dfrac{7}{24} \cdot \dfrac{24}{25} = \dfrac{7}{25}.

Tests that cover right triangle trig from two ratios:
SBACACTSATLEAP 2025NY State Math
More Grade 10 skills:
Perimeter and arc lengthSimilar right triangle trigonometryMidpoint formulaSpecial right trianglesEquation of a circleSurface area in context (SAT)Circle theoremsSurface area and volumeAll Grade 10 practice →

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