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Math101learn.math101.caCommon Trigonometry Errors
A rigorous diagnosis of triangle, unit-circle, identity, equation, inverse-function, and calculator-mode errors.
Precise definition
Trigonometry errors often come from choosing the wrong model: right-triangle ratios require a right triangle and acute reference context, while unit-circle definitions handle all real angles. Identities are equations true throughout a common domain; trigonometric equations seek particular solution sets.
Notation and mathematical language
On the unit circle, $(\cos\theta,\sin\theta)$ fixes signs by quadrant. $\tan\theta=\sin\theta/\cos\theta$ requires $\cos\theta\ne0$. Inverse notation $\sin^{-1}x$ means arcsine, not $1/\sin x$; reciprocal is $\csc x$.
Conceptual picture
Reference angles determine magnitudes and quadrants determine signs. Periodicity means an inverse-trig calculator returns a principal value, after which all solutions in the requested domain must be generated.
Conditions and key results
Angle units matter: calculus formulas use radians unless stated, and a calculator in the wrong mode produces plausible-looking wrong values. Dividing an equation by $\sin x$ or $\cos x$ can lose solutions where that factor is zero.
A reliable strategy
- Draw the triangle or unit-circle angle, mark quadrant, and confirm degree/radian mode.
- Choose a definition or identity whose conditions match the task.
- Solve algebraically, retaining zero-factor cases and generating periodic solutions.
- Substitute exact or numerical values and check the requested interval.
Fully worked example
Interpretation and application
Trigonometry models rotation, waves, surveying, and vectors. A sinusoidal fit to data describes periodic association; it does not prove the proposed mechanism causes the cycle.
Common mistakes
Verification and reasonableness
- Substitute every solution into the original equation.
- Check unit-circle coordinates and periodic copies.
- Estimate signs and magnitudes by quadrant before trusting a calculator.
Practice
- What is $\sin^{-1}(1/2)$ principal value in radians?
- Where is tangent undefined?
- Expand $\sin(a+b)$.
Answers and brief solutions
- $\pi/6$.
- Where $\cos x=0$.
- $\sin a\cos b+\cos a\sin b$.
Further deduction
Identity verification should transform one side at a time into the other or a common expression. Beginning by assuming the two sides are equal and performing reversible-looking steps can hide circular reasoning. Domain restrictions also matter: an identity involving tangent is asserted only where all displayed terms are defined.
The ambiguous SSA case explains why inverse sine alone can miss a triangle. Suppose sides $a=8$, $b=10$ and angle $A=30^\circ$ are given. The sine law gives $\sin B=b\sin A/a=10(1/2)/8=0.625$. One possibility is $B\approx38.68^\circ$, but sine has the same value at $180^\circ-B\approx141.32^\circ$. Both leave a positive third angle because $30^\circ+141.32^\circ<180^\circ$, so two triangles exist. A calculator's $\arcsin$ returns only the principal angle; it does not settle the geometry. Each candidate must satisfy the angle sum, positive-side conditions, and the side–angle ordering rule that the larger side faces the larger angle. This is the triangle analogue of finding all periodic solutions to a trigonometric equation rather than reporting one inverse-function output.
Identity manipulations must retain domains. Dividing by $\cos x$ converts an equation to tangent form only where $\cos x\ne0$ and can discard solutions at its zeros. Factoring before division preserves zero-factor cases for separate testing.
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
How many solutions does $\sin x(2\sin x-1)=0$ have on $[0,2\pi)$?
- $\sin x=0$ gives $0,\pi$.
- $\sin x=1/2$ gives $\pi/6,5\pi/6$, for four total.
End of lesson
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