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Trigonometry Ratios & Unit Circle cheat sheet - grade 9-11

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Math Grade 9-11

Trigonometry Ratios & Unit Circle Cheat Sheet

A printable reference covering sine, cosine, tangent, SOH-CAH-TOA, unit circle coordinates, radians, and reciprocal identities for grades 9-11.

TopicsTrigonometryRatios and ProportionsGeometryFunctionsAlgebraMathematics
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Trigonometry ratios connect angle measures to side lengths in right triangles and to coordinates on the unit circle. This cheat sheet helps students quickly choose the correct ratio, identify exact values, and move between degrees and radians. It is especially useful for solving triangle problems, graphing trig functions, and preparing for algebra, geometry, or precalculus assessments.

The core ratios are sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta, often remembered with SOH-CAH-TOA. On the unit circle, each point has coordinates (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta), which makes exact trig values easier to organize. Important ideas include reciprocal identities, quadrant signs, reference angles, and radian measure using 180∘=π180^{\circ} = \pi radians.

Key Facts

  • SOH-CAH-TOA means sin⁡θ=oppositehypotenuse\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, cos⁡θ=adjacenthypotenuse\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, and tan⁡θ=oppositeadjacent\tan \theta = \frac{\text{opposite}}{\text{adjacent}}.
  • The tangent ratio can also be written as tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} when cos⁡θ≠0\cos \theta \neq 0.
  • On the unit circle, the point at angle θ\theta is (x,y)=(cos⁡θ,sin⁡θ)(x,y) = (\cos \theta, \sin \theta).
  • The Pythagorean identity is sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.
  • The reciprocal identities are csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}, sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, and cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}.
  • To convert degrees to radians, multiply by π180∘\frac{\pi}{180^{\circ}}, so 60∘=π360^{\circ} = \frac{\pi}{3}.
  • To convert radians to degrees, multiply by 180∘π\frac{180^{\circ}}{\pi}, so 3π4=135∘\frac{3\pi}{4} = 135^{\circ}.
  • Quadrant signs follow ASTC: in Quadrant I all are positive, in Quadrant II sine is positive, in Quadrant III tangent is positive, and in Quadrant IV cosine is positive.

Vocabulary

Sine
Sine is the ratio sin⁡θ=oppositehypotenuse\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} in a right triangle and the yy-coordinate on the unit circle.
Cosine
Cosine is the ratio cos⁡θ=adjacenthypotenuse\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} in a right triangle and the xx-coordinate on the unit circle.
Tangent
Tangent is the ratio tan⁡θ=oppositeadjacent\tan \theta = \frac{\text{opposite}}{\text{adjacent}} and also equals sin⁡θcos⁡θ\frac{\sin \theta}{\cos \theta} when defined.
Unit Circle
The unit circle is a circle with radius 11 centered at the origin, where each angle θ\theta corresponds to (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta).
Reference Angle
A reference angle is the acute angle between the terminal side of an angle and the xx-axis.
Radian
A radian is an angle measure based on arc length, where 180∘=π180^{\circ} = \pi radians.

Common Mistakes to Avoid

  • Confusing opposite and adjacent sides is wrong because these sides depend on the chosen angle θ\theta, not on the picture alone.
  • Using tan⁡θ=adjacentopposite\tan \theta = \frac{\text{adjacent}}{\text{opposite}} is wrong because tangent is defined as tan⁡θ=oppositeadjacent\tan \theta = \frac{\text{opposite}}{\text{adjacent}}.
  • Swapping unit circle coordinates is wrong because the correct ordered pair is (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta), not (sin⁡θ,cos⁡θ)(\sin \theta, \cos \theta).
  • Forgetting quadrant signs gives incorrect exact values because a reference angle only gives the size of the value, while the quadrant determines whether it is positive or negative.
  • Treating degrees and radians as the same unit is wrong because 60∘60^{\circ} and 6060 radians are very different angle measures.

Practice Questions

  1. 1 In a right triangle, angle θ\theta has opposite side 99 and hypotenuse 1515. Find sin⁡θ\sin \theta and the adjacent side length.
  2. 2 Convert 150∘150^{\circ} to radians and find sin⁡150∘\sin 150^{\circ} using the unit circle.
  3. 3 Find cos⁡5π3\cos \frac{5\pi}{3}, sin⁡5π3\sin \frac{5\pi}{3}, and tan⁡5π3\tan \frac{5\pi}{3}.
  4. 4 Explain why sin⁡θ\sin \theta is positive but cos⁡θ\cos \theta is negative for an angle in Quadrant II.

Understanding Trigonometry Ratios & Unit Circle

A useful way to understand trigonometry is to focus on similarity. Two right triangles can have different sizes but the same sharp angle. Their matching side lengths change by the same scale factor.

This keeps each side comparison constant. That is why an angle can determine a ratio without knowing the triangle's actual size. In a word problem, first mark the angle being used.

Then label the sides relative to that angle. The hypotenuse is always across from the right angle.

Opposite and adjacent can change when you choose the other sharp angle. Many errors come from labeling sides before deciding which angle matters.

The unit circle extends this triangle idea beyond right triangle diagrams. Imagine a radius rotating from the positive horizontal axis. Its horizontal and vertical movements change smoothly as the angle turns.

The same values repeat after one complete rotation. This repeating behavior explains why sine and cosine graphs make waves. It explains why an angle larger than one full turn can have the same values as a smaller angle.

Angles that land at matching positions have matching coordinates. Negative angles rotate in the opposite direction. Watching the direction of rotation helps students place angles correctly before trying to recall any exact value.

Radians are important because they measure turning through arc length rather than through a named fraction of a circle. On a circle with radius one, the length of the arc swept out by an angle equals its radian measure. This makes radians a natural fit for circular motion and calculus.

A bicycle wheel, a clock hand, a fan blade, and a Ferris wheel all involve rotation. If a wheel has a known radius and turns through a known radian angle, its distance along the rim can be found directly from radius times angle. Degrees are often easier for describing familiar turns, while radians work more naturally in advanced formulas.

Exact values should be learned as patterns, not as a long list. The special angles come from two basic triangles, one with equal legs and one with side lengths related to one, square root of three, and two. Once the positive values are known, the quadrant tells whether each value is positive or negative.

Reference angles give the size of the acute angle between the terminal side and the horizontal axis. This reduces a difficult-looking angle to a familiar one. Check whether a result makes sense.

A sine or cosine value cannot be greater than one in size. Tangent becomes undefined when the horizontal coordinate is zero.

Reciprocal functions are undefined whenever the original ratio is zero. These restrictions matter when simplifying expressions, solving equations, and reading graphs.