Right-triangle problems connect side lengths and acute angles. Choose a relationship based on the sides given and requested, label opposite and adjacent relative to the specified angle, and reserve hypotenuse for the side opposite the right angle.
Essential concepts
\[sin heta= rac{ ext{opposite}}{ ext{hypotenuse}} qquad cos heta= rac{ ext{adjacent}}{ ext{hypotenuse}} qquad an heta= rac{ ext{opposite}}{ ext{adjacent}}\]
SOH-CAH-TOA summarizes the three ratios.
SAT strategy and application
Find a trigonometric ratio
A right triangle has legs 6 and 8 and hypotenuse 10. For the acute angle opposite the side of length 6, find its sine.
- Identify the ratio
Sine is opposite divided by hypotenuse.
- Substitute and simplify
\(sin heta=6/10=3/5\).
Sine: \( rac35\).
Mistakes and traps
Check your understanding
Relative to an acute angle, the opposite leg is 5 and the adjacent leg is 12. What is the tangent of the angle?
- \(5/12\)
- \(12/5\)
- \(5/13\)
- \(12/13\)
Show answer and explanation
Answer: \( rac5{12}\)
Tangent is opposite divided by adjacent.
Key takeaways
What to remember
- Label opposite and adjacent relative to the named angle.
- Use the Pythagorean theorem only with right triangles.
- Choose the trig ratio containing the known and requested sides.
- Check calculator angle mode when numerical angle measures are required.
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.