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Question
In ΔABC right angled at B, ∠A = ∠C. Find the value of:
(i) sinA cosC + cosA sinC
(ii) sinA sinB + cosA cosB
Solution
Since ∠B is right angled ⇒ ∠B = 90°
In ΔABC,
∠A + ∠B + ∠C = 180°
But ∠A = ∠C
⇒ ∠A + 90° + ∠A = 180°
⇒ 2∠A = 90°
⇒ ∠A = 45° = ∠C
(i) sinA cosC + cosA sinC
= sin45° cos45° + cos45° sin45°
= `(1)/sqrt(2) xx (1)/sqrt(2) + (1)/sqrt(2) xx (1)/sqrt(2)`
= `(1)/(2) + (1)/(2)`
= 1
(ii) sinA sinB + cosA cosB
sin45° sin90° + cos45° cos90°
= `(1)/sqrt(2) xx 1 + (1)/sqrt(2) xx 0`
= `(1)/sqrt(2)`.
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