Question

How sin30=1/2.explain in brief. And how sin^2+cos^2=1

Answer 1

Step-by-step explanation:

$\huge\sf\underline{answer}$

to solve :-

how sin30° = $\sf\dfrac{1}{2}$

how sin²A+cos²A = 1

$\huge\sf\underline{solution}$

1) to prove that sin 30° = 1

note :- please kindly refer to above attachment.

let ABC be an equilateral traingle whose each side is 2a. By geometry,each angle of the triangle is 60°. Let AD is.perpendicular to BC.

AD bisects \_BAC and it also bisects the side BC

.: angle CAD = angle BAD and CD = BD = a

in the right angle triangle ADC,

(AD)²+(DC)² = (AC)²

(AD)² + a² = (2a)²

(AD)² = (2a)²-a²

(AD)² = 4a²-a²

.: (AD)² = 3a²

AD = $\sf\sqrt{3}$ ×a

.: sin30° = $\sf\dfrac{CD}{AC}$

= $\sf\dfrac{a}{2a}$

= $\sf\dfrac{1}{2}$

.: sin 30° = $\sf\dfrac{1}{2}$

"hence proved"..

2) prove that sin²D+cos²D = 1

note:- kindly refer to above attachment.

consider a right angle triangle ABC, with angle B = 90° and angle CAB = D

from the diagram sinD = BC/AC

cosD = AB/AC

from Pythagoras theorem,

(AB)²+(BC)² = (AC)²

dividing both sides with AC² we get :-

(AB/AC)² + (BC/AC)² = 1

(cosD)²+(sin D)² = 1

.: sin²D + cos²D = 1

.: "hence proved"