Trigonometric Ratios
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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