Unit Circle Values
Trigonometry
Intro: We normalize the angle and return exact values (fractions & radicals) plus a decimal check.
Worked example
- θ = 5π/6.
- Normalize (already principal): $$\theta=\tfrac{5\pi}{6}\in(0,\pi).$$
- Reference angle: $$\alpha = \pi-\theta = \pi-\tfrac{5\pi}{6} = \tfrac{\pi}{6}.$$
- Exact values at the reference: $$\cos\alpha = \tfrac{\sqrt{3}}{2},\quad \sin\alpha = \tfrac{1}{2},\quad \tan\alpha = \tfrac{1}{\sqrt{3}}.$$
- Quadrant: $$\theta\in\text{QII}\;\Rightarrow\; \cos<0,\; \sin>0,\; \tan<0.$$
- Apply signs: $$\cos\theta = -\tfrac{\sqrt{3}}{2},\quad \sin\theta = \tfrac{1}{2},\quad \tan\theta = -\tfrac{1}{\sqrt{3}} = -\tfrac{\sqrt{3}}{3}.$$
- Coordinate on unit circle: $$(\cos\theta,\sin\theta)=\Big(-\tfrac{\sqrt{3}}{2},\tfrac{1}{2}\Big).$$
- Decimal check (≈): $$\cos\theta\approx-0.8660,\; \sin\theta\approx0.5000,\; \tan\theta\approx-0.57735.$$
- Answer: $$\boxed{(\cos\theta,\sin\theta,\tan\theta)=\Big(-\tfrac{\sqrt{3}}{2},\tfrac{1}{2},-\tfrac{\sqrt{3}}{3}\Big)}.$$
- θ = −135°.
- Normalize (to [0°,360°) if desired): $$-135^{\circ}+360^{\circ}=225^{\circ}.$$
- Reference angle: $$\alpha = 225^{\circ}-180^{\circ}=45^{\circ}.$$
- Exact reference values: $$\cos45^{\circ}=\sin45^{\circ}=\tfrac{\sqrt{2}}{2},\quad \tan45^{\circ}=1.$$
- Quadrant for 225° (QIII): $$\cos<0,\; \sin<0,\; \tan>0.$$
- Apply signs: $$\cos(-135^{\circ})=\cos(225^{\circ})=-\tfrac{\sqrt{2}}{2},\quad \sin(-135^{\circ})=\sin(225^{\circ})=-\tfrac{\sqrt{2}}{2},\quad \tan(-135^{\circ})=1.$$
- Coordinate: $$(\cos\theta,\sin\theta)=\Big(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2}\Big).$$
- Decimal check (≈): $$\cos\theta\approx-0.70711,\; \sin\theta\approx-0.70711,\; \tan\theta\approx1.$$
- Answer: $$\boxed{(\cos\theta,\sin\theta,\tan\theta)=\Big(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},1\Big)}.$$
FAQs
Cosecant/secant/cotangent?
Yes—reciprocals of sin, cos, tan respectively. We can return all six values on request.
How are signs chosen?
By quadrant: QI (+,+,+), QII (−,+,−), QIII (−,−,+), QIV (+,−,−) for (cos, sin, tan).
Non-special angles?
We provide decimal values and can rationalize where exact radicals exist.
Why choose Mathix?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
Unit circle values
- Reduce θ into a principal range (e.g., add/subtract multiples of 2π or 360°)
- Find α in {0,30,45,60,90,…} by reflecting to QI
- Use quadrant signs for cos, sin, tan; then provide exact radicals and a decimal check