Trigonometry Review

Precise definition

Trigonometry relates angles, rotation, and ratios. On the unit circle, $\cos\theta$ is the $x$-coordinate, $\sin\theta$ the $y$-coordinate, and $\tan\theta=\sin\theta/\cos\theta$ where $\cos\theta\ne0$. Radians measure arc length divided by radius.

Notation and mathematical language

Core identities include $\sin^2\theta+\cos^2\theta=1$, reciprocal and quotient identities, and angle-sum formulas. Sine and cosine have period $2\pi$; tangent has period $\pi$. In right triangles, ratios use acute reference angles and labelled sides.

Conceptual picture

The unit circle unifies triangle ratios with all real angles and determines quadrant signs. Graph transformations encode amplitude, period, phase shift, and vertical shift. Exact values expose structure that decimals conceal.

Conditions and key results

Identities hold only on their common domains. Equation solving requires all periodic solutions in the requested interval. The sine and cosine laws handle general triangles under their conditions; right-triangle formulas alone do not.

A reliable strategy

1. Convert units and draw the angle in standard position with a reference angle.
2. Use unit-circle coordinates or a labelled triangle to obtain exact ratios.
3. For identities, transform one side with valid laws; for equations, isolate a ratio and generate all quadrant/periodic solutions.
4. Check domain, interval, calculator mode, graph, and substitution.

Fully worked example

Interpretation and application

Trigonometry models waves, rotation, surveying, vectors, and periodic data. A sinusoidal model can predict within an observed regime, but a periodic correlation does not prove a causal mechanism.

Common mistakes

Verification and reasonableness

• Substitute exact solutions into the original equation.
• Use unit-circle signs and graph periods.
• Check triangle angle sum, side order, and a second ratio or theorem.

Practice

1. Convert $60^\circ$ to radians.
2. Find $\sin(5\pi/6)$.
3. State the period of $\tan x$.

Further deduction

A review should alternate four modes: exact unit-circle values, symbolic identities, equation solution sets, and applications. Add graph interpretation and radians before calculus. If one mode fails, repair the dependency—for example, quadrant signs or factoring—then return to a mixed problem so the skill is selected rather than cued.

For a general triangle, choose the cosine law when three sides or two sides with the included angle are known, and the sine law when a known side–opposite-angle pair is available. With sides 5 and 7 enclosing $60^\circ$, the third side satisfies $c^2=5^2+7^2-2(5)(7)\cos60^\circ=39$, so $c=\sqrt{39}$. The value is plausible because it lies between $|7-5|=2$ and $12$. Area can be found directly as $K=ab\sin C/2=35\sqrt3/4$. In equation work, identities transform expressions but do not remove the need for a complete interval-based solution. After finding a reference angle, enumerate quadrants consistent with the sign and add the correct period; then test endpoints according to whether the interval is open, closed, or half-open.

Keep angular units visible when technology is used. The same numeric input produces different values in degree and radian modes, and a correct symbolic formula can appear wrong under the wrong setting. Exact special-angle values provide a fast mode check.

Related topics

• Common Trigonometry Errors
• Right Triangle
• Unit Circle