Common Mistakes
Learn to recognize and avoid the most frequent errors students make in trigonometry, from calculator mishaps to conceptual misunderstandings.
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Understanding common mistakes helps you avoid them on tests and strengthens your conceptual foundation. This lesson covers the most frequent trigonometry errors and how to prevent them.
Category 1: Calculator and Mode Errors
Mistake: Wrong Angle Mode (Degrees vs. Radians)
The Problem: Your calculator is in degree mode when the problem uses radians, or vice versa.
Example: Calculating sin(pi) and getting 0.0548... instead of 0.
The Fix: Always check your calculator mode before starting. If working with pi, you need radian mode. If working with degree symbols (like 45 degrees), you need degree mode.
Quick Check: sin(30 degrees) should equal 0.5 in degree mode. sin(pi/6) should equal 0.5 in radian mode.
Mistake: Inverse Function Confusion
The Problem: Confusing sin^(-1)(x) (arcsin) with 1/sin(x) (cosecant).
Example: Thinking sin^(-1)(0.5) means 1/sin(0.5).
The Fix: Remember that sin^(-1) is the inverse function (arcsin), which gives you an angle. csc(x) = 1/sin(x) is the reciprocal function.
Category 2: Quadrant and Sign Errors
Mistake: Forgetting Quadrant Signs
The Problem: Finding the correct magnitude but wrong sign for a trig value.
Example: Finding sin(210 degrees) = 1/2 instead of -1/2.
The Fix: Use "All Students Take Calculus" (ASTC) to remember which functions are positive in each quadrant:
- Quadrant I: All positive
- Quadrant II: Sine positive
- Quadrant III: Tangent positive
- Quadrant IV: Cosine positive
Mistake: Reference Angle Errors
The Problem: Calculating the wrong reference angle.
Example: For 240 degrees, using 60 degrees reference angle correctly, but for 150 degrees, using 50 degrees instead of 30 degrees.
The Fix: Reference angle formulas:
- QI: theta
- QII: 180 - theta (or pi - theta)
- QIII: theta - 180 (or theta - pi)
- QIV: 360 - theta (or 2pi - theta)
Category 3: Identity Misapplication
Mistake: Incorrect Identity Formulas
The Problem: Misremembering or misapplying trigonometric identities.
Common Errors:
- Writing sin(A + B) = sin(A) + sin(B) (WRONG!)
- Writing cos^2(x) = cos(x^2) (WRONG!)
- Confusing sin^2(x) + cos^2(x) = 1 with tan^2(x) + cot^2(x) = 1 (the latter is wrong)
The Fix: Memorize the correct identities:
- sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
- cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
- sin^2(x) means (sin(x))^2
Mistake: Cancellation Errors
The Problem: Incorrectly canceling terms in fractions.
Example: Simplifying (sin(x) + cos(x))/sin(x) as 1 + cos(x) instead of 1 + cot(x).
The Fix: You can only cancel factors, not terms. Split the fraction first: (sin(x) + cos(x))/sin(x) = sin(x)/sin(x) + cos(x)/sin(x) = 1 + cot(x).
Category 4: Unit Circle Errors
Mistake: Mixing Up Coordinates
The Problem: Confusing which coordinate is cosine and which is sine.
Example: Saying sin(0) = 1 and cos(0) = 0.
The Fix: Remember: on the unit circle, the point is (cos(theta), sin(theta)). Cosine is always the x-coordinate (horizontal), sine is always the y-coordinate (vertical). At theta = 0, the point is (1, 0), so cos(0) = 1 and sin(0) = 0.
Mistake: Confusing Special Angle Values
The Problem: Mixing up the values for 30, 45, and 60 degree angles.
The Fix: Create a memory device. Notice that for sine, as the angle increases (30, 45, 60), the values increase: 1/2, sqrt(2)/2, sqrt(3)/2. For cosine, it's the reverse.
Examples
Identify the mistake in each problem and correct it.
Example 1: Find the Error
Student Work: Find cos(5pi/4).
5pi/4 is in Quadrant III. Reference angle is pi/4.
cos(pi/4) = sqrt(2)/2.
Answer: sqrt(2)/2
Error: The student forgot to apply the quadrant sign. In Quadrant III, cosine is negative.
Correct Answer: cos(5pi/4) = -sqrt(2)/2
Example 2: Find the Error
Student Work: Simplify sin(2x)/sin(x).
Cancel sin from top and bottom: 2x/x = 2
Answer: 2
Error: You cannot cancel sin as if it were a factor. sin(2x) is not equal to 2 * sin(x).
Correct Approach: Use the double angle identity: sin(2x) = 2sin(x)cos(x).
So sin(2x)/sin(x) = 2sin(x)cos(x)/sin(x) = 2cos(x)
Example 3: Find the Error
Student Work: If sin(theta) = 3/5, find cos(theta).
Using sin^2 + cos^2 = 1:
(3/5)^2 + cos^2 = 1
cos^2 = 1 - 9/25 = 16/25
cos = 4/5
Error: The student didn't consider that cos could be negative. Without knowing the quadrant, there are two possible answers.
Correct Answer: cos(theta) = +/- 4/5 (need quadrant information to determine the sign)
Practice
For each problem, identify whether there is an error. If there is, explain the mistake and provide the correct answer.
Problem 1: A student says tan(90 degrees) = 1 because "tangent at 45 degrees is 1, and 90 is twice 45."
Show Answer
Error: tan(90 degrees) is undefined because cos(90 degrees) = 0, and tan = sin/cos would involve division by zero. Trig functions are not linear.
Problem 2: sin(7pi/6) = sin(pi/6) = 1/2
Show Answer
Error: 7pi/6 is in Quadrant III where sine is negative. Correct answer: sin(7pi/6) = -1/2
Problem 3: cos(60 degrees) = sqrt(3)/2
Show Answer
Error: The values are swapped. cos(60 degrees) = 1/2. It's sin(60 degrees) that equals sqrt(3)/2.
Problem 4: To find sin^(-1)(0.5), I calculated 1/sin(0.5) = 2.086
Show Answer
Error: sin^(-1) means the inverse function (arcsin), not the reciprocal. sin^(-1)(0.5) = 30 degrees or pi/6 radians.
Problem 5: sec(pi/3) = 1/sin(pi/3) = 2/sqrt(3)
Show Answer
Error: Secant is the reciprocal of cosine, not sine. sec(pi/3) = 1/cos(pi/3) = 1/(1/2) = 2
Problem 6: (sin(x))^2 + (cos(x))^2 = 1, so sin(x) + cos(x) = 1
Show Answer
Error: You cannot take the square root of both sides of a sum like this. sqrt(a^2 + b^2) is not equal to a + b. The original identity is correct, but the conclusion is wrong.
Problem 7: The reference angle for 315 degrees is 315 - 270 = 45 degrees
Show Answer
Correct! 315 degrees is in Quadrant IV, so the reference angle is 360 - 315 = 45 degrees. (The student's method also works since 315 degrees is 45 degrees past the negative x-axis going clockwise.)
Problem 8: cot(x) = tan(1/x)
Show Answer
Error: Cotangent is 1/tan(x) or cos(x)/sin(x), not tan(1/x). These are completely different expressions.
Problem 9: sin(-theta) = sin(theta) because squaring makes negatives positive
Show Answer
Error: Sine is an odd function, meaning sin(-theta) = -sin(theta). You might be thinking of sin^2(-theta) = sin^2(theta), which is true.
Problem 10: In the equation y = 3sin(2x), the amplitude is 3 and the period is 2pi.
Show Answer
Error: The amplitude is correct (3), but the period is 2pi/2 = pi, not 2pi. The coefficient of x affects the period as period = 2pi/B where B is that coefficient.
Problem 11: cos(pi/2) = 1 because at pi/2 the point on the unit circle is at the top.
Show Answer
Error: At pi/2, the point is (0, 1). Since cosine is the x-coordinate, cos(pi/2) = 0. It's sin(pi/2) that equals 1.
Problem 12: If tan(theta) = 3/4 in Quadrant I, then sin(theta) = 3 and cos(theta) = 4.
Show Answer
Error: Sine and cosine values must be between -1 and 1. If tan = 3/4, we need to find the hypotenuse: sqrt(3^2 + 4^2) = 5. So sin = 3/5 and cos = 4/5.
Check Your Understanding
Test yourself on avoiding common mistakes.
Question 1: What should you always check on your calculator before solving trig problems?
Question 2: In which quadrants is tangent positive?
Question 3: What is the difference between sin^(-1)(x) and csc(x)?
Question 4: Why is sin(A + B) not equal to sin(A) + sin(B)?
Show Answers
- Check whether the calculator is in degree mode or radian mode.
- Quadrants I and III (where sine and cosine have the same sign)
- sin^(-1)(x) is the inverse sine function (arcsin) which returns an angle. csc(x) is the reciprocal, equal to 1/sin(x).
- Trigonometric functions are not linear/distributive. The correct formula is sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
Next Steps
- Create a personal "error log" to track mistakes you make
- Review problems you got wrong and identify the error type
- Take the Unit Quiz to test your overall understanding
- Double-check your calculator mode before every problem set