Trigonometric Ratios of Standard Angles

Trigonometric Ratios of Standard Angles

Trigonometric Ratios of Standard Angles

We have to remember the values of trigonometric ratios of the standard angles like 0°, 30°, 45°, 60°, 90° and so on. Those values can be derived and understood as follows:


Trigonometric ratios of 45°

Trigonometric ratios of 45°

Let OX be the initial line and trace out an XOP = 45°. From Q, a point on OP, draw QNOX. Then NQO = 90° - 45° = 45°. Therefore it is an isosceles triangle, i.e. ON = NQ. Let ON = NQ = a. Now, by Pythagoras relation,

          OQ2 = ON2 + NQ2

                 = a2 + a2

                 = 2a2

       ∴  OQ = a√2

Now for right angled triangle ONQ with angle of reference NOQ = 45°,

          Sin45° = NQ/OQ = a/a√2 = 1/√2

          Cosec45° = OQ/NQ = a√2/a = √2

          Cos45° = ON/OQ = a/a√2 = 1/√2

          Sec45° = OQ/ON = a√2/a = √2

          tan45° = NQ/ON = a/a = 1

          cot45° = ON/NQ = a/a = 1

 

Trigonometric ratios of 60° and 30°

Trigonometric ratios of 60° and 30°

Let ABC be a triangle in which AB = BC = CA and hence A = B = C = 60°. From A draw ADBC. Then BAD = CAD = 30° and BD = CD. Let AB = BC = CA = 2a. So, BD = CD = a.

Now, from right angled DABD,

          AD2 = AB2 – BD2

                 = 4a2 – a2

                 = 3a2

       ∴  AD = a√3

Now, in right angled DABD with angle of reference ABD = 60°,

          Sin60° = AD/AB = a√3/2a = √3/2

          Cosec60° = AB/AD = 2a/a√3 = 2/√3

          Cos60° = BD/AB = a/2a = 1/2

          Sec60° = AB/BD = 2a/a = 2

          tan60° = AD/BD = a√3/a = √3

          cot60° = BD/AD = a/a√3 = 1/√3

 And, with the angle of reference BAD = 30°,

          Sin30° = BD/AB = a/2a = 1/2

          Cosec30° = AB/BD = 2a/a = 2

          Cos30° = AD/AB = a√3/2a = √3/2

          Sec30° = AB/AD = 2a/a√3 = 2/√3

          tan30° = BD/AD = a/a√3 = 1/√3

          cot30° = AD/BD = a√3/a = √3

 

Trigonometric ratios of 0°

Let OX be the initial line and trace out an XOP = θ. Let Q be a point on OP. Draw QMOX.

Trigonometric ratios of 0°

When θ is small enough, QM will be small enough.

And, when θ = 0°, QM = 0 and OQ = OM. Now, from right angled triangle MOQ with angle of reference θ = 0°,

          Sin0° = QM/OQ = 0/OQ = 0

          Cosec0° = OQ/QM = OQ/0 = ∞ (undefined)

          Cos0° = OM/OQ = OM/OM = 1

          Sec0° = OQ/OM = OM/OM = 1

          tan0° = QM/OM = 0/OM = 0

          cot0° = OM/QM = OM/0 = ∞ (undefined)

 

Trigonometric ratios of 90°

Let OX be the initial line and trace out an XOP = θ. Let Q be a point on OP. Draw QMOX.

Trigonometric ratios of 90°

When MOQ = θ is close to 90°, OM will be close to 0.

And, when θ = 90°, OM = 0 and OQ = MQ. Now, from right angled triangle MOQ with angle of reference θ = 90°,

          Sin90° = MQ/OQ = MQ/MQ = 1

          Cosec90° = OQ/MQ = MQ/MQ = 1

          Cos90° = OM/OQ = 0/OM = 0

          Sec90° = OQ/OM = OQ/0 = ∞ (undefined)

          tan90° = MQ/OM = MQ/0 = ∞ (undefined)

          cot90° = OM/MQ = 0/MQ = 0


From the above relations, the values of various trigonometric ratios of standard angles can be summarized as follows:

Table of trigonometric ratios of standard angles: 0°, 30°, 45°, 60° and  90°.

 

Worked Out Examples

Example 1: Find the value of sin230° + sin245° + sin260° Solution: Here, 	sin230° + sin245° + sin260° 	= (1/2)^2+ (1/√2)^2+ (√3/2)^2 = 1/4+ 1/2+ 3/4 = (1 + 2 + 3)/4 = 6/4 = 3/2


Example 2: Find the value of tan 〖π/3〗^csin 〖π/3〗^c+ sin 〖π/4〗^ccos 〖π/2〗^c+ cos 〖π/2〗^csin 〖π/3〗^c Solution: Here, 	tan 〖π/3〗^csin 〖π/3〗^c+ sin 〖π/4〗^ccos 〖π/2〗^c+ cos 〖π/2〗^csin 〖π/3〗^c 	= tan60°sin60° + sin45°cos90° + cos90°sin60° = √3 × √3/2 + 1/√2 × 0 + 0 × √3/2   = 3/2  + 0 + 0   = 3/2


Example 3: Prove that: (1 + tan30°)/(1 - tan30°) = (cos30° + sin30°)/(cos30° - sin30°) Solution: Here,     L.H.S. = (1 + tan30°)/(1 - tan30°) 	  = (1 + 1/√3)/(1- 1/√3)     = ( (√3  + 1)/√3)/((√3- 1)/√3)     = (√3  + 1)/(√3- 1)    R.H.S. = (cos30° + sin30°)/(cos30° - sin30°) 	  = (√3/2  + 1/2)/(√3/2  + 1/2)     = ( (√3  + 1)/√3)/((√3- 1)/√3)     = (√3  + 1)/(√3- 1) ∴ L.H.S. = R.H.S.  Proved.


Example 4: Find the value of x: sin60°cos30° + x sin45°cos45° = sin30°cos60° Solution: Here, 	sin60°cos30° + x sin45°cos45° = sin30°cos60° or,	√3/2  × √3/2  + x 1/√2 × 1/√2 = 1/2 × 1/2 or,	3/4 + x/2 = 1/4 or,	x/2 = 1/4  - 3/4 or,	x/2 = (1 - 3)/4 or,	x/2 = (- 2)/4 or,	x/2 = (- 1)/2 or,	x = -1 ∴  x = -1

 

Example 5: In the given figure, the distance between house B and House C is 20m and BCA = 30°. Calculate the distance between house A and B.

Example 5: In the given figure, the distance between house B and House C is 20m and ∠BCA = 30°. Calculate the distance between house A and B.

Solution:

          As triangle ABC is a right angled triangle and BC = 20m, BCA = 30° and BAC = 90°.

          Sin30° = AB/BC

or,     ½ = AB/20m

or,     2AB = 20m

or,     AB = 20m/2

or,     AB = 10m

Hence, the distance between house A and B is 10m.

 

You can comment your questions or problems regarding the values of trigonometric ratios of standard angles here.

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