Question

In the figure, ABC is a right angled triangle, B= 90°....Write down all the trigonometric ratios of A and C​

Answer 1

Given :-

A right angled triangle ABC, right angled at B.

To Find :-

All the trigonometric ratios of C and A

Solution :-

We know that,

$\sf \leadsto\red{sin \: θ = \frac{perpendicular}{hypotenuse} } \: \: \: \: \\ \sf \leadsto \orange{cos \: θ = \frac{base}{hypotenuse} } \: \: \: \: \: \: \: \\ \sf \leadsto \pink{tan \: θ = \frac{perpendicular}{base} } \: \: \: \\ \sf \leadsto \green{cosec \: θ = \frac{hypotenuse}{perpendicular} } \\ \sf \leadsto \gray{sec \: θ = \frac{hypotenuse}{base} } \: \: \: \: \: \: \: \: \\ \sf \leadsto \blue{cot \: θ = \frac{base}{perpendicular} } \: \: \: \:$

Now, when observer i.e. θ is A,

• Perpendicular = BC
• Base = AB
• Hypotenuse = AC

Therefore,

$\sf \leadsto\red{sin \: A = \frac{BC}{AC} } \: \: \: \\ \sf \leadsto \orange{cos \: A = \frac{AB}{AC} } \: \: \\ \sf \leadsto \pink{tan \: A = \frac{BC}{AB} } \: \: \: \\ \sf \leadsto \green{cosec \: A = \frac{AC}{BC} } \\ \sf \leadsto \gray{sec \: A = \frac{AC}{AB} } \: \: \: \\ \sf \leadsto \blue{cot \: A = \frac{AB}{BC} } \: \:$

Now, when observer i.e θ is C,

• Base = BC
• Perpendicular = AB
• Hypotenuse = AC

Therefore,

$\sf \leadsto\red{sin \: C = \frac{AB}{AC} } \: \: \: \: \\ \sf \leadsto \orange{cos \: C = \frac{BC}{AC} } \: \: \: \: \\ \sf \leadsto \pink{tan \: C = \frac{AB}{BC} } \: \: \: \\ \sf \leadsto \green{cosec \: C = \frac{AC}{AB} } \\ \sf \leadsto \gray{sec \: C = \frac{AC}{BC} } \: \: \: \: \\ \sf \leadsto \blue{cot \: C = \frac{BC}{AB} } \: \: \: \:$

Hope it helps you :)

Answer 2

Question -

In the figure, ABC is a right angled triangle, B= 90°....Write down all the trigonometric ratios of A and C

Answer -

Here ∠CAB is an acute angle. Observe the position of the sides with respect to angle A.

- BC is the opposite side of Angle A

- AB is the adjacent side with respect to angle A.

- AC is the hypotenuse of the right angled triangle ABC

The trigonometric ratios of the angle A in the right angled triangle ABC can be defined as follows.

Sin A - Side Opposite To angle A / Hypotenes = BC/AC

cos A - Side Adjacent to Angle A / Hypotenes = AB/AC

tan A - Side opposite To Angle A / Side Adjacent to Angle A = BC/AB

csc A - Hypotheses / Side opposite To Angle A = AC/BC

sec A - Hypotheses / Side Adjacent to Angle A =AC/AB

cot A - side adjecent To Angle A / Side Opposite To angle A = AC/BC

Now let us define the trigonometric ratios for the acute angle C in the right angled triangle,

∠ABC= 90°

Observe that the position of the sides changes when we consider angle 'C' in place of angle A.

• Sin C - AB / AC
• cos C - BC / AC
• tan C - AB / BC
• csc C - AC / AB
• Sec C - AC / BC
• cot C - BC / AB

Hope it's Helpful.

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