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Question
Form the following figure, find the values of:
- cos B
- tan C
- sin2B + cos2B
- sin B. cos C + cos B. sin C
Solution
Given angle BAC = 90°
⇒ BC2 = AB2 + AC2 ...(BC is hypotenuse)
⇒ 172 = 82 + AC2
⇒ AC2 = 289 - 64
⇒ AC2 = 225
⇒ AC = `sqrt225`
∴ AC = 15
(i) cos B = `"base"/"hypotenuse" = "AB"/"BC" = 8/17`
(ii) tan C = `"perpendicular"/"base" = "AB"/"AC" = 8/15`
(iiii) sin B = `"perpendicular"/"hypotenuse" = "AC"/"BC" = 15/17`
cos B = `"base"/"hypotenuse" = "AB"/"BC" = 8/17`
∴ sin2 B+ cos2 B = `(("perpendicular")/("hypotenuse"))^2 + (("base")/("hypotenuse"))^2`
= ` (15/ 17)^2 + (8 /17)^2`
= `(225 + 64) / (289) `
= `(289)/(289) `
= 1
(iv) sin B = `"perpendicular"/"hypotenuse" = "AC"/"BC" = 15/17`
cos B = `"base"/"hypotenuse" = "AB"/"BC" = 8/17`
sin C = `"perpendicular"/"hypotenuse" = "AB"/"BC" = 8/17`
cos C = `"base"/"hypotenuse" = "AC"/"BC" = 15/17`
sin B · cos C + cos B · sin C
= ` 15/17. 15/17 + 8/17. 8/17 `
= `( 225 + 64 )/ (289)`
= `(289)/(289)`
= 1
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