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Question
Without using tables, evaluate the following: (sin90° + sin45° + sin30°)(sin90° - cos45° + cos60°).
Solution
(sin90° + sin45° + sin30°)(sin90° - cos45° + cos60°).
sin30° = `(1)/(2)`
sin45° = `(1)/sqrt(2)`
sin90° = 1
cos45° = `(1)/sqrt(2)`
cos60° = `(1)/(2)`
(sin90° + sin45° + sin30°)(sin90° - cos45° + cos60°)
= `(1 + 1/sqrt(2) + 1/sqrt(2))(1 - 1/sqrt(2) + 1/sqrt(2))`
= `(3/2 + 1/sqrt(2))(3/2 - 1/sqrt(2))`
= `(3/2)^2 - (1/sqrt(2))^2`
= `(9)/(4) - (1)/(2)`
= `(9 - 2)/(4)`
= `(7)/(4)`.
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