 # Application of Surds to Trigonometric Ratios

Welcome to class!

In today’s class, we will be talking about the application of surds to trigonometric ratios. Enjoy the class!

### Application of surds to trigonometric ratios Apart from solving quadratics, surds also arise in trigonometry.

The angles 30°, 45°, 60° have the following trigonometric ratios.

Triangle ABC is equilateral. AD is the line interval from A to the midpoint of BD. Triangles ABD and ACD are congruent (SSS).

Hence,

 sin 30° = cos 60° = 1/2 , sin 60° = cos 30° = $\surd 3/2$  , tan 30° = $1/\surd 3$   and tan 60° = $\surd 3$  .

#### The trigonometric functions of y = cos θ and y = cos θ.

 θ° 0 30 60 90 120 150 180 210 240 270 300 330 360 sin θ 0 0.5 0.87 1 0.87 0.5 0 −0.5 −0.87 −1 −0.87 −0.5 0

More points can be used to show that the shape of the graph is as shown below. Electrical engineers and physicists call this a sine wave. The basic formulae here all reply on the SINE graph as follows:

• The sine of an angle is defined by the vertical height of a point as it rotates around a unit circle (that is, its radius is 1) measured from a horizontal line through the centre of the circle. So it cannot be bigger than 1 or less than -1.
• the cosine of an angle is defined by the horizontal distance of a point as it rotates around the unit circle measured from a vertical line through the centre of the circle. It too must be in the range -1 to 1.

#### The surd form of trigonometric ratios: The equilateral triangle above has sides each 2 units long and all angles at 60o. It has been halved into 2 right – angled triangles of base 1 unit long. Using Pythagoras:

(ab)2 = (ac)2 + (bc)2
=> (2)2 = (ac)2 + (1)2
=> (ac)2 = 4 – 1 = 3
and (ac) = root 3.

Using the definitions of trigonometric ratios, summarised in SOHCAHTOA:

sin 30o = bc / ab = 1 / 2
sin 60o = ac / ab = root 3 / 2
cos 30o = ac / ab = root 3 / 2
cos 60o = bc / ab = 1 /2
tan 30o = bc / ac = 1 / root 3
tan 60 = ac / bc = root 3 / 1 = root 3 The isosceles above has equal sides of 1 unit each and the 2 complementary angles each of 45o. Using Pythagoras’ Theorem to find side (ac):

(ac)2 = (ab)2 + (bc)2
(ac)2 = (1)2 + (1)2 = 2
ac = root 2

 Trigonometrical ratio: Surd form: Approximation sin 30o 1 / 2 0.5 sin 45o 1 / root 2 0.7071 sin 60o root 3 / 2 0.866 cos 30o root 3 / 2 0.866 cos 45 1 / root 2 0.7071 cos 60o 1 / 2 0.5 tan 30o 1 / root 3 0.5774 tan 45o 1 tan 60o root 3 1.7321
##### For any pair of complementary angles these 2 rules apply:

sin x = cos (90 – x)
and cos x = sin (90 – x)

 sin 30o = cos 60o sin 60o = cos 30o sin 45o = cos 45o sin 20o = cos 70o cos 41o = sin 49o cos 72o = sin 18o sin 36o = cos 54o When the angle is 30o on a right-angled triangle, then the side opposite the 30o angle is half the hypotenuse. This is also true for similar triangles. Sides H, A and O can be any value and O = 1 / 2 H. Therefore, sin 30o = 0.5.

In our next class, we will be talking about Matrices and Determinant. We hope you enjoyed the class.

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