Question

A unit circle is shown below with centre $O$ . A tangent $AB$ is drawn to the circle at point $M$ such that $∠ MOB = β$

(Note : the figure is not to scale)

If $OA ⊥ OB$ , write the expression that represents the length of

(i) $OB$

(ii) $OA$

(iii) $AB$

Accepted Answer

Given that a unit circle with centre $O$ , $AB$ is a tangent to the circle at point $M$ such that $∠ MOB = β$ , and $OA ⊥ OB$ as shown below-

$∠ OMB = ∠ OMA = 90 °$ (The radius and tangent of a circle at a point of contact is always perpendicular to each other)

(i) So, by applying trigonometric ratios in right-angled $∆ OMB$ -

$\cos (β) = \frac{OM}{OB} = \frac{1}{OB} \Rightarrow O B = \frac{1}{\cos β}$

$\Rightarrow OB = \sec β$

(ii) $∵ OA ⊥ OB \Rightarrow ∠ AOB = 90 ° \Rightarrow ∠ AOM = 90 ° - β$

So, by applying trigonometric ratios in $∆ OMA$ -

$\cos (90 ° - β) = \frac{OM}{OA} = \frac{1}{OA} \Rightarrow sinβ = \frac{1}{OA} \Rightarrow OA = \frac{1}{sinβ}$

$\Rightarrow OA = cosec β$

iii) Now, by applying Pythagoras' theorem in right-angled $∆ AOB$

${AB}^{2} = {OA}^{2} + {OB}^{2}$

$\Rightarrow {AB}^{2} = {cosec}^{2} β + \sec^{2} β = \frac{1}{\sin^{2} β} + \frac{1}{\cos^{2} β}$

$\Rightarrow {AB}^{2} = \frac{\cos^{2} β + \sin^{2} β}{\sin^{2} {βcos}^{2} β}$

$\Rightarrow AB = \sqrt{\frac{1}{\sin^{2} {βcos}^{2} β}}$

$\Rightarrow AB = \frac{1}{sinβcosβ}$

Hence, $OB = \sec β, OA = cosec β$ and $AB = \frac{1}{sinβcosβ}$ .

A unit circle is shown below with centre $O$ . A tangent $AB$ is drawn to the circle at point $M$ such that $∠ MOB = β$

(Note : the figure is not to scale)

If $OA ⊥ OB$ , write the expression that represents the length of

(i) $OB$

(ii) $OA$

(iii) $AB$

Important Questions on Introduction to Trigonometry

Learn and Prepare for any exam you want !

A unit circle is shown below with centre O. A tangent AB is drawn to the circle at point M such that ∠MOB=β (Note : the figure is not to scale) If OA⊥OB, write the expression that represents the length of (i) OB (ii) OA (iii) AB

Important Questions on Introduction to Trigonometry

Learn and Prepare for any exam you want !

A unit circle is shown below with centre $O$ . A tangent $AB$ is drawn to the circle at point $M$ such that $∠ MOB = β$

(Note : the figure is not to scale)

If $OA ⊥ OB$ , write the expression that represents the length of

(i) $OB$

(ii) $OA$

(iii) $AB$