A reliable method
Label opposite, adjacent and hypotenuse before choosing sine, cosine or tangent. A stated ratio can form a proportion without calculating an angle.
The common trap
Opposite and adjacent change when you switch the reference angle. The hypotenuse does not. Never choose the longest-looking side without a right-angle fact.
Make the connection
Check a triangle ratio against reasonable bounds: sine and cosine of an acute angle lie between zero and one, while tangent need not.
Worked example: choose the requested quantity
In a right triangle, the side opposite angle A is 9 and the adjacent leg is 12. What is tan A? Tangent compares opposite to adjacent, not to hypotenuse. Divide the two stated leg lengths and reduce. The requested value is 3/4.
Contrast and diagnose an error
A right triangle has a leg of 12 adjacent to angle B and hypotenuse 15. What is cos B? Cosine compares adjacent leg to hypotenuse. The common scale factor cancels from the ratio. The requested value is 4/5. A tempting answer is 3/5: Use the opposite leg: sine rather than cosine.
Transfer practice and check
A right triangle has a 30° angle and hypotenuse 8. What is the length opposite 30°? Try it before reading on. Check: 4. The sine of 30 degrees is one half. Multiply the hypotenuse by that ratio.
In a right triangle, the side opposite angle A is 9 and the adjacent leg is 12. What is tan A?
- Tangent compares opposite to adjacent, not to hypotenuse.
- Divide the two stated leg lengths and reduce.
- The requested value is 3/4.
Compare the other answer choices
Use sine instead of tangent.
Use cosine instead of tangent.
Reverse opposite and adjacent.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content