Sin, Cos & Tan

Unit circle foundation

• Unit circle means a circle centered at the origin with radius $1$, so every point on it satisfies $x^2 + y^2 = 1$. This fixed radius turns trigonometry into a coordinate system rather than a triangle-only topic. It applies to all real angles because rotation can continue beyond $360^\circ$ (or $2\pi$).
• Coordinate definition: for an angle $\theta$ measured from the positive $x$-axis, the point on the unit circle is $(\cos\theta, \sin\theta)$. This works because the adjacent side over hypotenuse becomes $x/1$, and opposite over hypotenuse becomes $y/1$. So sine and cosine are literally coordinates.

Tangent as a ratio and slope

• Tangent is defined by $\tan\theta = \frac{\sin\theta}{\cos\theta}$ when $\cos\theta \neq 0$. This ratio is also the gradient of the line from the origin to the point $(\cos\theta, \sin\theta)$ on the unit circle. The slope meaning helps explain why tangent changes sign by quadrant and why it is undefined where $x=0$.

Why the definitions work

• Right-triangle logic extends to all angles by embedding a right triangle inside the unit circle. The radius acts as hypotenuse $1$, so ratios become direct coordinates and remain consistent under rotation. This is why the definitions are valid even for negative angles and angles greater than one revolution.
• Pythagorean identity follows from the circle equation, giving $\sin^2\theta + \cos^2\theta = 1$. This identity is a structural constraint, not a memorized trick, and it lets you check whether computed sine and cosine values are plausible.

Key Identity: $\sin^2\theta + \cos^2\theta = 1$

Periodicity and symmetry

• Periodicity means trig outputs repeat because rotating by a full cycle returns to the same point on the circle. Therefore, $\sin(\theta + 2\pi k)=\sin\theta$ and $\cos(\theta + 2\pi k)=\cos\theta$, while $\tan(\theta + \pi k)=\tan\theta$. This principle generates infinitely many solutions to trig equations.
• Symmetry explains related angles efficiently: sine is odd, cosine is even, and tangent is odd. Concretely, $\sin(-\theta)=-\sin\theta$, $\cos(-\theta)=\cos\theta$, and $\tan(-\theta)=-\tan\theta$. These relationships reduce calculation load and improve sign accuracy.

Solving basic trig equations in an interval

• Step 1: get a reference angle by inverse trig, such as $\alpha=\sin^{-1}(k)$ or $\alpha=\cos^{-1}(k)$. This gives a principal angle but not the complete answer set. You then use geometry of the unit circle to locate other angles with the same coordinate property.
• Step 2: place all matching quadrants using sign rules and coordinate matching. For sine, match the same $y$-value; for cosine, match the same $x$-value; for tangent, match the same slope and use opposite quadrants. This converts one calculator value into all required solutions in the stated interval.
• Step 3: apply periodic extensions by adding or subtracting $360^\circ$ (or $2\pi$) for sine/cosine, and $180^\circ$ (or $\pi$) for tangent. This ensures no valid solution in the target interval is missed. Always list answers in order and include interval endpoints when valid.

Building exact values and related angles

• Exact values are values expressed using fractions and surds, such as $\sin 45^\circ=\frac{\sqrt{2}}{2}$. They come from special triangles and unit-circle coordinates, so they are conceptual facts rather than calculator approximations. They are especially useful in proofs and non-calculator work.
• Related-angle method uses a first-quadrant reference angle and quadrant sign. For example, if the reference angle is $\alpha$, then an angle in quadrant II is typically $\pi-\alpha$, and in quadrant III it is $\pi+\alpha$. This method is faster and less error-prone than recomputing from scratch.

Function behavior and domain differences

• Sine and cosine are bounded, so their outputs stay between $-1$ and $1$, while tangent is unbounded and can take any real value. This matters when checking whether an equation like $\sin x = 1.4$ is possible, because it is immediately impossible. By contrast, $\tan x = 1.4$ is always feasible.
• Undefined points for tangent occur where $\cos\theta=0$, so $\tan\theta$ has vertical asymptotes there. This is a domain restriction, not a graphing artifact, and it must be respected in equations and transformations.

Degrees vs radians and primary vs complete solutions

• Angle units are not interchangeable in calculations, so mode consistency is essential. If the interval uses multiples of $\pi$, work in radians throughout to avoid conversion mistakes. Mixed-unit work is a common source of correct method but wrong final answers.
• Primary value is only a starting point, not the final solution set. Inverse trig returns one angle by convention, but periodicity and symmetry create additional solutions. Complete solving means combining principal value, quadrant logic, and interval filtering.

Typical errors and why they happen

• Mistaking reference angle for actual angle causes quadrant sign errors. Students often compute a positive acute angle and forget to reflect it into the correct quadrant required by the equation. The fix is to separate reference-angle size from quadrant
• Assuming tangent exists everywhere leads to invalid values at odd multiples of $\frac{\pi}{2}$. This misconception appears when learners treat tangent like sine and cosine. Always check whether $\cos\theta=0$ before using $\tan\theta=\frac{\sin\theta}{\cos\theta}$.

Precision and representation mistakes

• Over-rounding early can shift final angles and make boundary decisions wrong. Keep exact forms or sufficient decimal precision until the final requested rounding stage. This is especially important when intervals are tight.
• Rote memorization without geometry makes CAST and exact values fragile under stress. When memory fades, coordinate meaning and symmetry reconstruct the facts reliably. Conceptual anchors outperform isolated lists.