Grade: Grade 10 Subject: Mathematics Unit: Trigonometry Basics Lesson: 5 of 6 SAT: Geometry+Trigonometry ACT: Math

Common Mistakes

Learn to recognize and avoid the most frequent errors students make when solving trigonometry problems.

Why Study Mistakes?

Understanding common errors helps you:

  • Catch your own mistakes before submitting answers
  • Develop stronger problem-solving habits
  • Build confidence by knowing what to watch out for
  • Improve your test-taking speed by avoiding common traps

Common Mistakes and How to Avoid Them

Mistake #1: Calculator in Wrong Mode

The Error: Getting a strange answer like sin(30) = -0.988 instead of 0.5
The Fix: Make sure your calculator is in DEGREE mode, not RADIAN mode. Look for "DEG" on your calculator display. Most trigonometry at this level uses degrees.
Quick Check: sin(30) should equal 0.5, and cos(60) should equal 0.5. If not, switch modes!

Mistake #2: Confusing Opposite and Adjacent

The Error: Mixing up which side is opposite and which is adjacent to the angle.
The Fix: Always identify your reference angle first. The opposite side is directly across from your angle (not touching it). The adjacent side touches your angle (but is not the hypotenuse).
Remember: Opposite and adjacent depend on which angle you're working with. The same side can be opposite to one angle and adjacent to another!

Mistake #3: Using the Wrong Ratio

The Error: Using sine when you should use tangent, or cosine when you should use sine.
The Fix: Before calculating, identify what you KNOW and what you NEED:
  • Know opposite, need hypotenuse (or vice versa): Use SINE
  • Know adjacent, need hypotenuse (or vice versa): Use COSINE
  • Know opposite, need adjacent (or vice versa): Use TANGENT
SOH-CAH-TOA: Write it at the top of every problem until it becomes automatic!

Mistake #4: Forgetting to Use Inverse Functions

The Error: Writing "angle = sin(0.5)" when you mean to find the angle whose sine is 0.5.
The Fix: When finding an angle from a ratio, you must use the inverse function: sin^(-1), cos^(-1), or tan^(-1). On most calculators, this is the "2nd" or "shift" button followed by sin, cos, or tan.
Notation: sin^(-1)(0.5) = 30 degrees. This is also written as arcsin(0.5) = 30 degrees.

Mistake #5: Setting Up the Ratio Upside Down

The Error: Writing sin(A) = hypotenuse/opposite instead of opposite/hypotenuse.
The Fix: Memorize the correct order. In SOH-CAH-TOA:
  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent
The hypotenuse is always in the denominator for sine and cosine!

Mistake #6: Not Checking if the Answer Makes Sense

The Error: Getting an answer like "the ladder is 500 feet long" when the problem says it reaches a 10-foot wall.
The Fix: Always do a reasonableness check:
  • The hypotenuse must be the longest side
  • Angles in a triangle must sum to 180 degrees
  • Acute angles in a right triangle must each be less than 90 degrees
  • Trig ratios for acute angles: sine and cosine are between 0 and 1

Mistake #7: Mixing Up Angle of Elevation and Depression

The Error: Using angle of depression incorrectly in calculations, leading to wrong triangle setup.
The Fix: Remember that angle of depression (looking down) equals the angle of elevation (looking up) from the other point due to alternate interior angles with a horizontal line. Draw the horizontal reference line at the observer's position.

Mistake #8: Rounding Too Early

The Error: Rounding intermediate steps, leading to significant error in the final answer.
The Fix: Keep full calculator precision throughout your calculations. Only round your final answer, typically to 2 decimal places unless otherwise specified.
Example: If tan(37) = 0.75355..., use the full value until the very end. Don't round to 0.75 mid-calculation.

Spot the Error

Find and correct the mistake in each problem.

Problem 1

Student Work: To find the height of a tree, I measured 50 feet from the base and found the angle of elevation to be 40 degrees. I calculated: height = 50 / tan(40) = 59.6 feet.

Find the Error

Error: The student divided instead of multiplied. The height is opposite and 50 feet is adjacent, so height = 50 * tan(40) = 42 feet, not 50/tan(40).

Problem 2

Student Work: In a right triangle with hypotenuse 13 and one leg 5, I found the angle opposite the leg of 5: angle = sin(5/13) = 0.0067 degrees.

Find the Error

Error: The student forgot to use inverse sine. The correct calculation is angle = sin^(-1)(5/13) = sin^(-1)(0.3846) = 22.62 degrees.

Problem 3

Student Work: To find side b in a triangle where angle B = 55 degrees and the hypotenuse = 20: I used cos(55) = b/20, so b = 20 * cos(55) = 11.47.

Find the Error

Error: If b is the side opposite angle B, then the student should use sine, not cosine. sin(55) = b/20, so b = 20 * sin(55) = 16.38. (Note: If the student meant b to be adjacent to angle B, then the work is correct.)

Problem 4

Student Work: sin(30) = 0.988 (I calculated this on my calculator)

Find the Error

Error: The calculator is in radian mode. sin(30 degrees) = 0.5, not 0.988. The student needs to switch to degree mode.

Problem 5

Student Work: In right triangle ABC with right angle at C, angle A = 35 degrees. I need to find angle B. Since A + B + C = 180, we have 35 + B + 90 = 180, so B = 180 - 35 - 90 = 55 degrees.

Find the Error

No Error: This work is actually correct! The two acute angles in a right triangle are complementary (sum to 90 degrees), and this calculation confirms that.

Problem 6

Student Work: A 20-foot ladder leans against a wall at a 70-degree angle with the ground. Height reached = 20 * cos(70) = 6.84 feet.

Find the Error

Error: The height is opposite the 70-degree angle, so use sine, not cosine. height = 20 * sin(70) = 18.79 feet. Cosine would give the distance from the wall to the base of the ladder.

Problem 7

Student Work: tan(A) = adjacent/opposite = 8/6, so A = tan^(-1)(8/6) = 53.13 degrees.

Find the Error

Error: Tangent is opposite/adjacent, not adjacent/opposite. The correct setup is tan(A) = 6/8, so A = tan^(-1)(0.75) = 36.87 degrees.

Problem 8

Student Work: I found that the hypotenuse of my triangle is 5 cm, and one leg is 12 cm. The other leg must be 13 cm by the Pythagorean theorem.

Find the Error

Error: The hypotenuse must be the longest side. A hypotenuse of 5 cm cannot have a leg of 12 cm. Either the student mislabeled the sides or made an error identifying the hypotenuse.

Problem 9

Student Work: From the top of a 100-meter cliff, the angle of depression to a boat is 25 degrees. I drew my triangle with the 25-degree angle at the boat and found the distance.

Find the Error

Error: The angle of depression is measured from the horizontal at the top of the cliff, not at the boat. However, this equals the angle of elevation from the boat (alternate interior angles), so if the student used this correctly in calculations, the final answer may still be right. The diagram description is what's incorrect.

Problem 10

Student Work: If sin(A) = 0.8, then A = 53.13 degrees. Therefore, cos(A) = cos(53.13) = 0.6.

Find the Error

No Error: This is correct! Using the Pythagorean identity: cos(A) = sqrt(1 - sin^2(A)) = sqrt(1 - 0.64) = sqrt(0.36) = 0.6. The student verified this by calculating cos(53.13 degrees).

Check Your Understanding

  1. What is the first thing you should check if your trigonometry answer seems completely wrong?
  2. How do you remember the correct order for SOH-CAH-TOA ratios?
  3. When finding an angle, what function type must you use?
  4. What quick checks can you do to verify your answer is reasonable?
Check Your Answers
  1. Check if your calculator is in degree mode (not radian mode).
  2. Use the mnemonic SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Write it at the top of your paper!
  3. You must use an inverse function (sin^(-1), cos^(-1), or tan^(-1)).
  4. Check that: the hypotenuse is the longest side, angles sum to 180 degrees, acute angles are between 0 and 90 degrees, and sine/cosine values are between 0 and 1 for acute angles.

Next Steps

  • Review any mistake types that you commonly make
  • Create a personal checklist of things to verify before submitting answers
  • When ready, take the Unit Quiz to test your mastery
  • Keep a log of your errors on practice tests to identify patterns