Trigonometric f.2

Without using a calculator.find the value of the following expressions ,
tan²60°cos²30°

tan²60°cos²30°
= (√3)² x [(√3)/2]²
= 3 x (3/4)
= 9/4
= 2.25


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In each of the following . a trigonometric ratio isgiven . Find the other 2 trigonometric ratios by constructing a right-angledtriangle
(1) sinθ = (√17)/9
(2) tanθ = 3

(1)
Construct a right-angled triangle with an angle θ. The opposite side of θ is √17,and the hypotenuse is 9.
The adjacent side
= √[(9)² - (√17)²] (Pythagorean theorem)
= 8

Hence,
cosθ = 8/9
tanθ = (√17)/8

(2)
Construct a right-angled triangle with an angle θ. The opposite side is 3 andthe adjacent side is 1.
The hypotenuse
= √[(3)² + (1)²] (Pythagorean theorem)
= √10

Hence,
sinθ = 3/√10 = 3(√10)/10
cosθ = 1/√10 = (√10)/10


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Simplify the following expressions.
(1) tanθ / sinθ - sin³θ
(2) cos²θ + cos²θ tan²θ

(1)
Case 1 :
tanθ / (sinθ - sin³θ)
= (sinθ / cosθ) / sinθ (1 - sin²θ)
= sinθ / [sinθ cosθ (1 - sin²θ)]
= 1 / (cosθ cos²θ)
= 1 / cos³θ

Case2 :
(tanθ / sinθ) - sin³θ
= [(sinθ / cosθ) / sinθ] - sin³θ
= (1 / cosθ) - sin³θ
= (1 - sin³θcosθ) / cosθ

(2)
cos²θ + cos²θtan²θ
= cos²θ (1 + tan²θ)
= cos²θ [1 + (sin²θ/cos²θ)
= cos²θ [(cos²θ + sin²θ)/cos²θ]
= cos²θ (1/cos²θ)
= 1