Trigonometric Ratios

Sine, Cosine & Tangent in Right Triangles

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Trigonometric ratios describe how the sides of a right triangle relate to one of its acute angles. They are essential in geometry, physics, engineering, and many real world measurement problems. With sine, cosine, and tangent, you can find missing sides or angles without measuring everything directly. These ratios turn triangle shapes into powerful tools for calculation.

In a right triangle, the names opposite, adjacent, and hypotenuse depend on which acute angle you choose. Sine compares opposite to hypotenuse, cosine compares adjacent to hypotenuse, and tangent compares opposite to adjacent. These relationships stay the same for all similar right triangles with the same angle. That is why trigonometric ratios let you solve many problems involving height, distance, slope, and direction.

Key Facts

sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent
hypotenuse^2 = opposite^2 + adjacent^2
sin(θ)/cos(θ) = tan(θ)
For any acute angle in a right triangle, 0 < sin(θ) < 1 and 0 < cos(θ) < 1

Vocabulary

Right triangle: A triangle that has one angle equal to 90 degrees.
Hypotenuse: The hypotenuse is the longest side of a right triangle and lies opposite the right angle.
Opposite side: The opposite side is the side across from the chosen angle θ.
Adjacent side: The adjacent side is the side next to the chosen angle θ that is not the hypotenuse.
Trigonometric ratio: A trigonometric ratio is a comparison of two side lengths in a right triangle relative to a chosen angle.

Common Mistakes to Avoid

Mixing up opposite and adjacent, because these labels change depending on which acute angle is chosen. Always identify the reference angle θ first before naming the sides.
Calling the longest side adjacent, which is wrong because the hypotenuse is always opposite the right angle. Find the 90 degree angle first, then label the side across from it as the hypotenuse.
Using tangent with the hypotenuse, which is wrong because tan(θ) compares only opposite and adjacent. Use tan(θ) = opposite/adjacent, not opposite/hypotenuse or adjacent/hypotenuse.
Applying trigonometric ratios to a triangle that is not right, which is wrong for these basic definitions. First confirm the triangle has a 90 degree angle before using SOHCAHTOA.

Practice Questions

1 In a right triangle, relative to angle θ, the opposite side is 8 cm and the hypotenuse is 10 cm. Find sin(θ).
2 In a right triangle, relative to angle θ, the adjacent side is 12 m and the opposite side is 5 m. Find tan(θ) and then use the Pythagorean theorem to find the hypotenuse.
3 A student says that in the same right triangle, a side can be opposite for one acute angle and adjacent for the other acute angle. Explain why this is correct.