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Trigonometric ratios connect the angles of a right triangle to the lengths of its sides. They are essential in geometry, physics, engineering, and any situation where you need to find a missing side or angle. Sine, cosine, and tangent let you turn shape information into equations you can solve. Once you know how to label opposite, adjacent, and hypotenuse, many triangle problems become systematic.
For a chosen acute angle in a right triangle, each trig ratio compares specific sides. Sine compares opposite to hypotenuse, cosine compares adjacent to hypotenuse, and tangent compares opposite to adjacent. These ratios depend only on the angle, not on the triangle's overall size, so similar triangles have the same trig values for matching angles. This makes trig a powerful tool for measurement, navigation, and modeling real-world situations.
Key Facts
- sin(theta) = opposite/hypotenuse
- cos(theta) = adjacent/hypotenuse
- tan(theta) = opposite/adjacent
- hypotenuse is always the side opposite the 90 degree angle
- tan(theta) = sin(theta)/cos(theta)
- For any right triangle, a^2 + b^2 = c^2 where c is the hypotenuse
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 90 degree angle.
- Opposite side
- For a chosen acute angle, the opposite side is the side directly across from that angle.
- Adjacent side
- For a chosen acute angle, the adjacent side is the side next to the angle that is not the hypotenuse.
- Trigonometric ratio
- A trigonometric ratio is a comparison of two side lengths in a right triangle based on a chosen angle.
Common Mistakes to Avoid
- Mixing up opposite and adjacent, because these names depend on the chosen acute angle. Always identify the reference angle first, then label sides relative to that angle.
- Calling the longest side adjacent, which is wrong because the hypotenuse is always opposite the 90 degree angle. Find the right angle first so you can identify the hypotenuse correctly.
- Using sine, cosine, or tangent with the wrong pair of sides, which leads to incorrect equations. Match the ratio to the side names before substituting numbers.
- Applying right triangle trig to a triangle that is not a right triangle, which is invalid for these basic definitions. Check that the triangle has a 90 degree angle before using SOH CAH TOA.
Practice Questions
- 1 In a right triangle, the side opposite angle theta is 6 cm and the hypotenuse is 10 cm. Find sin(theta) and cos(theta).
- 2 A right triangle has an acute angle of 35 degrees and an adjacent side of length 12 m. Write an equation using tangent or cosine to find the missing hypotenuse, then compute the hypotenuse to the nearest tenth.
- 3 Two different right triangles each have a 40 degree angle. Explain why their sine values for that angle are the same even if the triangles have different side lengths.