5.2 Trigonometric Functions of Real Numbers
Ideas in this lesson
- Coordinate definitions and domains
- Exact circular evaluation
- Signs and quadrant inference
- Diagram versus calculator evaluation
- Pythagorean identities and sign choice
- One trigonometric expression in another
- Parity of composite expressions
- Model substitution with units
- Geometric reduction formulas
- Angles versus real inputs
Coordinate definitions and domains · Worked example
Define all six trigonometric functions using a unit-circle terminal point. Give their global domains, then complete the six-function rows at \(2 \pi\), \(\dfrac{17 \pi}{6}\). Use column order sine, cosine, tangent, cosecant, secant, cotangent, and explain every undefined entry. Show the requested reasoning; written work and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: \(0\), \(1\), \(0\), undefined, \(1\), undefined, \(\dfrac{1}{2}\), \(- \dfrac{\sqrt{3}}{2}\), \(- \dfrac{\sqrt{3}}{3}\), \(2\), \(- \dfrac{2 \sqrt{3}}{3}\), \(- \sqrt{3}\)
\(\sin t=y,\ \cos t=x,\ \tan t=\dfrac{y}{x},\ \csc t=\dfrac1y,\ \sec t=\dfrac1x,\ \cot t=\dfrac{x}{y}\).
Sine and cosine allow all real inputs. Tangent and secant exclude \(t=\dfrac\pi2+k\pi\); cotangent and cosecant exclude \(t=k\pi\), for integers \(k\). A zero numerator gives zero only when the denominator is nonzero.
\(t=2 \pi\) gives \(\left(1,0\right)\).
\(t=\dfrac{17 \pi}{6}\) gives \(\left(- \dfrac{\sqrt{3}}{2},\dfrac{1}{2}\right)\).
Written-work review:
Sine and cosine allow all real inputs. Tangent and secant exclude \(t=\dfrac\pi2+k\pi\); cotangent and cosecant exclude \(t=k\pi\), for integers \(k\). A zero numerator gives zero only when the denominator is nonzero.
\(t=2 \pi\) gives \(\left(1,0\right)\).
| Sine | Cosine | Tangent | Cosecant | Secant | Cotangent |
|---|---|---|---|---|---|
| \(0\) | \(1\) | \(0\) | undefined | \(1\) | undefined |
\(t=\dfrac{17 \pi}{6}\) gives \(\left(- \dfrac{\sqrt{3}}{2},\dfrac{1}{2}\right)\).
| Sine | Cosine | Tangent | Cosecant | Secant | Cotangent |
|---|---|---|---|---|---|
| \(\dfrac{1}{2}\) | \(- \dfrac{\sqrt{3}}{2}\) | \(- \dfrac{\sqrt{3}}{3}\) | \(2\) | \(- \dfrac{2 \sqrt{3}}{3}\) | \(- \sqrt{3}\) |
Written-work review:
- Use the stated definitions and the complete allowed domain.
- Show all intermediate values and justify signs, restrictions and conclusions.
- Give every requested result in the stated order and units.