Trigonometrical Ratios of 90°

How to Find the Trigonometrical Ratios of 90°?

Let a rotating line  $\overrightarrow{OX}$ rotates about O in the anti-clockwise sense and starting from its initial position $\overrightarrow{OX}$ traces out ∠XOY = θ where θ is very nearly equal to 90°.

Let $\overrightarrow{OX}$ ⊥ $\overrightarrow{OZ}$ therefore, ∠XOZ = 90°

Take a point P on $\overrightarrow{OY}$ and draw $\overline{PQ}$ perpendicular to $\overline{OX}$.

Then,

Sin θ = $\frac{\overline{PQ}}{\overline{OP}}$;

cos θ = $\frac{\overline{OQ}}{\overline{OP}}$

and tan θ =$\frac{\overline{PQ}}{\overline{OQ}}$

When θ is slowly approaches 90° and finally tends to 90° then,

(a) $\overline{OQ}$ slowly decreases and finally tends to zero and

(b) the numerical difference between $\overline{OP}$  and $\overline{PQ}$  becomes very small and finally tends to zero.

Hence, in the Limit when θ → 90° then $\overline{OQ}$ → 0 and $\overline{PQ}$   → $\overline{OP}$  . Therefore, we get

$\lim_{θ \rightarrow 90°}$ sin  θ

= $\lim_{θ \rightarrow 90°}\frac{\overline{PQ}}{\overline{OP}}$

= $\frac{\overline{OP}}{\overline{OP}}$ [since, θ → 90° therefore, $\overline{PQ}$   → $\overline{OP}$ ].

= 1

Therefore sin 90° = 1

$\lim_{θ \rightarrow 90°}$ cos θ

= $\lim_{θ \rightarrow 90°}\frac{\overline{OQ}}{\overline{OP}}$

= $\frac{0}{\overline{OP}}$, [since, θ → 0° therefore, $\overline{OQ}$ → 0].

= 0

Therefore cos 90° = 0

$\lim_{θ \rightarrow 90°}$ tan θ

= $\lim_{θ \rightarrow 90°}\frac{\overline{PQ}}{\overline{OQ}}$

= $\frac{\overline{OP}}{0}$ [since, θ → 0° $\overline{OQ}$ → 0 and $\overline{PQ}$   → $\overline{OP}$].

= undefined

Therefore tan 900 = undefined

Thus,

csc 90° = $\frac{1}{sin  90°}$

= $\frac{1}{1}$, [since, sin 90° = 1] 

= 1

sec 90° = $\frac{1}{cos  90°}$

= $\frac{1}{0}$, [since, cos  90° = 0] 

= undefined

cot 0° = $\frac{ cos  90°}{ sin  90°}$

= $\frac{0}{1}$, [since, sin 900 = 1 and cos 90° = 0] 

= 0

Trigonometrical Ratios of 90 degree 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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