Use the trigonometric ratios sin, cos, and tan in right-angled triangles to find unknown sides and angles, including setting up the correct ratio for a given problem
Every right triangle has one 90 degree corner. The side across from it is the hypotenuse, always the longest. Pick one of the two acute angles. The side across from your angle is the opposite. The side touching your angle that is not the hypotenuse is the adjacent. Switch to the other acute angle and opposite and adjacent swap. The hypotenuse never changes.
Three ratios connect your angle to its sides. Sin is opposite over hypotenuse. Cos is adjacent over hypotenuse. Tan is opposite over adjacent. SOH CAH TOA keeps them in order. All right triangles with the same acute angle are similar, so these ratios depend only on the angle, not the triangle's size.
To find a missing side, pick the ratio naming the side you want and a side you know. If the unknown sits on top, multiply: opposite = hypotenuse times sin(angle). If the unknown sits underneath, divide: hypotenuse = adjacent ÷ cos(angle). To find a missing angle, form the ratio then undo it with the inverse, sin⁻¹, cos⁻¹ or tan⁻¹. The inverse is not a reciprocal. Set your calculator to degree mode first.
A tree stands vertical on level ground. You stand 20 metres from its foot and sight the top along a straight line from ground level at a 45 degree angle. The ground distance is adjacent, the tree height is opposite, so tan links them. Since tan(45°) = 1, height = 20 × 1 = 20 metres.
A ramp meets level ground at 60 degrees and its base is 6 metres. The base is adjacent, the ramp length is the unknown hypotenuse, so cos links them: cos(60°) = 6 ÷ length. Since cos(60°) = 0.5, length = 6 ÷ 0.5 = 12 metres. The unknown sat underneath, so we divide.
A right triangle has an opposite side of 6 m and a hypotenuse of 10 m. To find the angle, form sin θ = 6/10 = 0.6, then undo sin with its inverse: θ = sin⁻¹(0.6) ≈ 36.9°. The inverse is not 1/0.6. Set your calculator to degree mode first.
Match your known and wanted sides to one ratio before computing. The unknown's position decides the final step: on top means multiply, underneath means divide. For an angle, build the ratio from two known sides, then apply the matching inverse. Tan is the direct ratio when the two sides in play are opposite and adjacent, but sin or cos can find an unknown hypotenuse by division.
Name the sides from your chosen acute angle, pick the ratio naming the wanted and known sides, then multiply or divide by the unknown's position. Use an inverse function in degree mode to find a missing angle.
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Every right triangle has a hypotenuse across from the right angle. From a chosen acute angle, the side across is the opposite. Two sides touch your angle: the hypotenuse and the adjacent. The adjacent is the touching side that is not the hypotenuse. You stand 20 metres from a tree on level ground and sight the top along a straight line from ground level at a 45 degree angle.
The tree stands vertical. Relative to your 45 degree angle, which side is its height?
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