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### The trigonometric ratios

If we fix an acute angle ( heta), then all right-angled triangles that have ( heta) as one of their angles are similar. So, in all such triangles, corresponding pairs of sides are in the same ratio.

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The side opposite the right angle is called the **hypotenuse**. We label the side opposite ( heta) as the **opposite** and the remaining side as the **adjacent**. Using these names we can list the following standard ratios:

egin{align*}sin heta &= dfrac{ ext{opposite}}{ ext{hypotenuse}}, & cos heta &= dfrac{ ext{adjacent}}{ ext{hypotenuse}}, & an heta &= dfrac{ ext{opposite}}{ ext{adjacent}}.end{align*}

Allied to these are the three reciprocal ratios, **cosecant**, **secant** and **cotangent**:

egin{align*}operatorname{cosec} heta &= dfrac{ ext{hypotenuse}}{ ext{opposite}}, & sec heta &= dfrac{ ext{hypotenuse}}{ ext{adjacent}}, & cot heta &= dfrac{ ext{adjacent}}{ ext{opposite}}.end{align*}

These are called the reciprocal ratios as

egin{align*}operatorname{cosec} heta &= dfrac{1}{sin heta}, & sec heta &= dfrac{1}{cos heta}, & cot heta &= dfrac{1}{ an heta}.end{align*}

The trigonometric ratios can be used to find lengths and angles in right-angled triangles.

Example

Find the value of (x) in the following triangle.

Solution

We have

Hence

Special angles

The trigonometric ratios of the angles (30^circ), (45^circ) and (60^circ) can be easily expressed as fractions or surds, and students should commit these to memory.

Trigonometric ratios of special angles ( heta) (sin heta) (cos heta) ( an heta)(30^circ) | (dfrac{1}{2}) | (dfrac{sqrt 3}{2}) | (dfrac{1}{sqrt 3}) |

(45^circ) | (dfrac{1}{sqrt 2}) | (dfrac{1}{sqrt 2}) | 1 |

(60^circ) | (dfrac{sqrt 3}{2}) | (dfrac{1}{2}) | (sqrt 3) |

Triangles for the trigonometric ratios of special angles.Detailed description of diagram

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Extending the angles

In the module Further trigonometry (Year 10), we showed how to redefine the trigonometric functions in terms of the coordinates of points on the unit circle. This allows the definition of the trigonometric functions to be extended to the second quadrant.

Detailed description of diagram

If the angle ( heta) belongs to the first quadrant, then the coordinates of the point (P) on the unit circle shown in the diagram are simply ((cos heta, sin heta)).

Thus, if ( heta) is the angle between (OP) and the positive (x)-axis:

the **cosine** of ( heta) is defined to be the (x)-coordinate of the point (P) on the unit circlethe **sine** of ( heta) is defined to be the (y)-coordinate of the point (P) on the unit circle.

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We can apply this definition to any angle.

The **tangent** ratio is then defined by

< an heta = dfrac{sin heta}{cos heta},>provided (cos heta) is not zero.

As an example, let us take ( heta) to be (30^circ), so (P) has coordinates ((cos 30^circ, sin 30^circ)). Now move the point (P) around the circle to (P”), so that (OP”) makes an angle of (150^circ) with the positive (x)-axis. Note that (30^circ) and (150^circ) are supplementary angles. The coordinates of (P”) are ((cos 150^circ, sin 150^circ)).

Detailed description of diagram

But we can see that the triangles (OPQ) and (OP”Q”) are congruent, so the (y)-coordinates of (P) and (P”) are the same. Thus, (sin 150^circ = sin 30^circ). Also, the (x)-coordinates of (P) and (P”) have the same magnitude but opposite sign, so (cos 150^circ = -cos 30^circ).

From this typical example, we see that if ( heta) is any obtuse angle, then its supplement ({180^circ – heta}) is acute, and the sine of ( heta) is given by

Exercise 1

On the unit circle, place the point (P) corresponding to each of the angles (0^circ), (90^circ), (180^circ), (270^circ) and (360^circ). By considering the coordinates of each of these points, complete the following table of trigonometric ratios.

Trigonometric ratios of special angles Coordinates of (P) ( heta) (sin heta) (cos heta) ( an heta)(0^circ) | ||||

(90^circ) | undefined | |||

(180^circ) | ||||

(270^circ) | undefined | |||

(360^circ) |

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