Trigonometric Ratios
How to trigonometric ratios, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
Trigonometric Ratios: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Right-triangle sine
- Sine is opposite over hypotenuse.
- Substitute 5 and 13.
- Leave the ratio simplified.
Unit-circle cosine
- At angle 0, the point is (1, 0).
- Cosine is the x-coordinate.
- Read the x-value.
Try one before moving on
Trigonometric Ratios: pop-up practice
What are trigonometric ratios?
Sine: In the given triangle, the (sin) of the angle (θ) can be considered as follows, (color{blue}{sin: θ = frac{AB}{AC}}).
Cosine: In the given triangle, the (cos) of the angle (θ) can be considered as follows, (color{blue}{cos: θ = frac{BC}{AC}}).
Tangent: In the given triangle, the (tan) of the angle (θ) can be considered as follows, (color{blue}{tan: θ = frac{AB}{BC}}).
Cosecant: In the given triangle, the (cosec) of the angle (θ) can be considered as follows, (color{blue}{cosec: θ = frac{AC}{AB}}).
Secant: In the given triangle, the (sec) of the angle (θ) can be considered as follows, (color{blue}{sec: θ = frac{AC}{BC}}).
Cotangent: In the given triangle, the (cot) of the angle (θ) can be considered as follows, (color{blue}{cot: θ = frac{BC}{AB}}).
Trigonometric ratios formulas
We can use the shorthand form of trigonometric ratios to compare the length of both sides with the base angle. The angle (θ) is acute ((θ<90º)) and in general is measured with reference to the positive (x)-axis, in the anticlockwise direction. The basic trigonometric ratio formulas are given below,
- (color{blue}{sin: θ = frac{Perpendicular}{Hypotenuse}})
- (color{blue}{cos: θ = frac{Base}{Hypotenuse}})
- (color{blue}{tan: θ = frac{Perpendicular}{Base}})
- (color{blue}{sec: θ =frac{Hypotenuse}{Base}})
- (color{blue}{cosec: θ = frac{Hypotenuse}{Perpendicular}})
- (color{blue}{cot: θ = frac{Base}{Perpendicular}})
Trigonometric Ratios, Example 1:
Find the value of (tan:θ) if (sin:θ:=frac{10}{3}) and (cos:θ:=frac{5}{3}:).
Solution: Use the formula of the trigonometric ratio to solve this problem: (tan: θ = frac{Perpendicular}{Base}).
(tan:θ =frac {10}{5}=2)
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