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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