Trigonometrical Ratios of 30°

How to find the trigonometrical Ratios of 30°?

Let a rotating line $\overrightarrow{OX}$ rotates about O in the anti-clockwise sense and starting from the initial position $\overrightarrow{OX}$ traces out ∠XOY = 30°.

Take a point P on $\overrightarrow{OY}$ and draw PA perpendicular to $\overrightarrow{OX}$ Then, ∠OPA = 60°.

Now, produce PA to B such that PA = MB and join OB.

From ∆PMO and ∆QMO we have,

PA = BA,

OA common

and ∠OBP = ∠OPB = 60°

Therefore, ∠POB = 30° + 30° = 60°; which shows that each angel of triangle OPQ is 60° . Hence ∆OPQ is equilateral.

Let, OP = PB = 2a; therefore, PA = ½ PB = a

Again, OA² + PA² = OP²

⇒ OA² + a² = (2a)²

⇒ OA² = 4a² – a²

⇒ OA² = 3a²

Therefore, OA = √3a (Since, OA > 0).

Now, from the right-angled ∆OPA we have,

sin 30° = $\frac{\overline{PA}}{\overline{OP}} = \frac{a}{2a} = \frac{1}{2}$;

cos 30° = $\frac{\overline{OA}}{\overline{OP}} =  \frac{\sqrt{3}a}{2a} = \frac{\sqrt{3}}{2}$

And tan 30°  = $\frac{PA}{OA} = \frac{a}{\sqrt{3}a} = \frac{1}{\sqrt3} = \frac{\sqrt{3}}{3}$

Therefore, csc 30° = $\frac{1}{sin  30°}$  = 2;

Sec 30° = $\frac{1}{cos  30°} = \frac{2}{\sqrt3} = \frac{2\sqrt{3}}{3}$

And cot 30° = $\frac{1}{tan  30°}$ =  √3.

Trigonometrical Ratios of 30° are commonly called standard angles and the trigonometrical ratios of these angles are frequently used to solve particular angles.

● Trigonometric Functions

• Basic Trigonometric Ratios and Their Names
• Restrictions of Trigonometrical Ratios
• Reciprocal Relations of Trigonometric Ratios
• Quotient Relations of Trigonometric Ratios
  
• Limit of Trigonometric Ratios
  
• Trigonometrical Identity
• Problems on Trigonometric Identities
• Elimination of Trigonometric Ratios
• Eliminate Theta between the equations
• Problems on Eliminate Theta
• Trig Ratio Problems
• Proving Trigonometric Ratios
• Trig Ratios Proving Problems
• Verify Trigonometric Identities
• Trigonometrical Ratios of 0°
• Trigonometrical Ratios of 30°
• Trigonometrical Ratios of 45°
• Trigonometrical Ratios of 60°
• Trigonometrical Ratios of 90°
• Trigonometrical Ratios Table
• Problems on Trigonometric Ratio of Standard Angle
  
• Trigonometrical Ratios of Complementary Angles
• Rules of Trigonometric Signs
• Signs of Trigonometrical Ratios
  
• All Sin Tan Cos Rule
• Trigonometrical Ratios of (- θ)
• Trigonometrical Ratios of (90° + θ)
• Trigonometrical Ratios of (90° - θ)
• Trigonometrical Ratios of (180° + θ)
• Trigonometrical Ratios of (180° - θ)
• Trigonometrical Ratios of (270° + θ)
• Trigonometrical Ratios of (270° - θ)
• Trigonometrical Ratios of (360° + θ)
• Trigonometrical Ratios of (360° - θ)
• Trigonometrical Ratios of any Angle
• Trigonometrical Ratios of some Particular Angles
• Trigonometric Ratios of an Angle
• Trigonometric Functions of any Angles
• Problems on Trigonometric Ratios of an Angle
• Problems on Signs of Trigonometrical Ratios

11 and 12 Grade Math

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