Trigonometric Ratio Of Special Angles And Logical Reasoning SS1 Further Mathematics Lesson Note

The basic trigonometric ratios can be defined in terms of the sides of a right-angled triangle.

 For example, ▲PQR is a right-angle triangle with QPR = Ө and PRQ = 90˚

We define the three basic ratios as follows:

Cosine of angle Ө = PR    =  q

                      PQ         r    

Sine of angle Ө = QR    =      p

                                PQ              r

Tangent of angle Ө = QR    =     p         

            PR           r      

The cosine of angle Ө, sine of angle Ө and the tangent of angle Ө will be abbreviated as cosӨ, sinӨ and tanӨ respectively.

Thus: cosӨ = q ,sinӨ  = p, tanӨ = p

Also, sinӨ  =   p/r  =   p     =  tanӨ

            q/r          q   

tanӨ =  sinӨ

               cosӨ

Reciprocals of Basic Ratios

We define the reciprocals of the three basic ratios as:

The secant of angle Ө =   PQ/PR    = r/q   = 1 / cosine of angle Ө.

Cosecant of angle Ө = PQ / QR = r / q = 1 / sine of angle Ө 

Cotangent of angle Ө = PR / QR   = q / p =   1 / tangent of angle Ө 

The secant of angle Ө, the cosecant of angle Ө and the cotangent of angle Ө are abbreviated secӨ, cosecӨ and cotӨ respectively.

SecӨ   = r / q   = 1 / cosӨ

CosecӨ   = r / p =   1 / sinӨ

CotӨ =   q / p =   1 / tanӨ   =   cosӨ / sinӨ

Example 1

Given that sinӨ = 5 / 13 and Ө is acute, find:

1. cosӨ
2. tanӨ
3. secӨ
4. cosecӨ
5. cotӨ

Solution

Use Pythagoras’ theorem to find PR

PQ2 = PR2 + QR2

132 =PR2 + 52

PR2 = 132 – 52

      = 169 – 25

      = 144

PR = 12

Thus, q = 12, r = 13, p = 5.

cosӨ = q / r = 12 /13

tanӨ = p / q = 5 / 12

secӨ = r / q = 13 / 12

cosecӨ = r / p = 13 / 5

cotӨ = q / p = 12 / 5 

Ratios of General Angle

First Quadrant

sinӨ = y

cosӨ= x

tanӨ = y / x 

Example: Use the table to evaluate (a) sin37 (b) cos75 (c) tan62

Solution

a) sin37 = 0.6018

b) cos75 = 0.2588

c) tan62 = 1.881

Second Quadrant

Sin (180 – Ө) = sinӨ

Cos (180 – Ө) = -cosӨ

Tan (180 – Ө) = -tanӨ

Example: Use the table to evaluate (a) sin143 (b) cos 115 (c) tan 125

Solution

sin143 = sin(180-143) = sin37 = 0.6018

cos115 = -cos(180-115) = -cos65 = -0.4226

tan125 = -tan(180-125) = -tan55 = -1.428

Third Quadrant

Sin (180 + Ө) =   – sinӨ

Cos (180 + Ө) =   – cosӨ

Tan (180 + Ө) =     tanӨ

Example: Use table to evaluate (a) sin220 (b) cos236 (c) tan242

Solution

sin220 =   sin (180 + 40) =  –  sin40 =  – 0.6428

cos236 =   cos (180 + 56) = – cos56 =  – 0.5992

tan242 =   tan (180 + 62) =   tan 62 =   1.881

Fourth Quadrant

Sin (360 – Ө) =   – sinӨ

Cos (360 – Ө) =    cosӨ

Tan (360 – Ө) =   – tanӨ

Example: Use the table to evaluate (a) sin3100 (b) cos2850 (c) tan3340

Solution

sin3100 = – sin (360-310) =  – sin50 = – 0.7660

cos2850 =   cos (360-285)  =  cos75 =    0.2588

tan3340 = – tan (360-334) =  – tan26 =  – 0.4877

Note that:       

1. In the first quadrant, all the ratios are positive.
2. In the second quadrant, only the sine ratio is positive, while the rest are negative.
3. In the third quadrant, only the tangent ratio is positive, while the rest are negative.
4. In the fourth quadrant, only the cosine ratio is positive, while the rest are negative.