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Illustration 1:Since in general, it is simpler to use the value of 30°, so we will have to consider the triangle with 30°. Here,
Find the angles and sides indicated by the letters in the diagram. Give each answer correct to the nearest whole number.
r/60 = tan 30°.
Hence, r = 60tan (30°) = 34.64101615...
So answer correct to the nearest whole number is r= 35.
Now, that we have obtained the value of r, we use the first triangle to evaluate s. r/s = tan(55°)
so, 35/s = tan(55°)
so, 35/tan(55°) = s = 24.50726384...
r = 35, s = 25
Evaluate the value of (1 + cos π/8) (1 + cos 3π/8) (1 + cos 5π/8) (1 + cos 7π/8).
Solution: The given expression is
(1 + cos π/8) (1 + cos 3π/8) (1 + cos 5π/8) (1 + cos 7π/8)
= (1 + cos π/8) (1 + cos 3π/8) (1 - cos 3π/8) (1 - cos π/8)
= (1 – cos2π/8) (1 - cos23π/8)
= ¼ [2 sin π/8 sin 3π/8]2
= ¼ [2 sin π/8 cos π/8]2
= ¼ [sin π/4]2
= 1/4. 1/2 = 1/8
Illustration 3: If cos (α – β) = 1 and cos (α + β) = 1/e, where α, β ∈ [-π, π], then the values of α and β satisfying both the equations is/are
Solution: It is given in the question that cos (α – β) = 1 and cos (α + β) = 1/e, where α, β ∈ [-π, π].
Now, cos (α – β) = 1
α – β = 0, 2π, -2π
Hence, α – β = 0 (since α, β ∈ [-π, π])
So, α = β
Hence, cos 2α = 1/e
So, the number of solutions of above will be number of points of intersection of the curves y = cos 2α and y = 1/e
where α, β ∈ [-π, π]
It is quite clear that there are four solutions corresponding to four points of intersection P1, P2, P3 and P4.
Illustration 4: If k = sin (π/18) sin (5π/18) sin (7π/18), then what is the numerical value of k?
Solution: The value of k is given to be k = sin (π/18) sin (5π/18) sin (7π/18).
Hence, k = sin 10° sin 50° sin 70°
= sin 10° sin (60° - 10°). sin (60° + 10°)
= sin 10° [sin260° - sin210°]
= sin 10° [(√3/2)2 - sin210°]
= sin 10° [3/4 - sin210°]
= 1/4 [3sin 10° - 4sin310°]
= 1/4 x sin (3 x 10) (since, sin 3θ = 3sinθ- 4sin3 θ)
= 1/4 sin 30° = 1/8
Hence, the numerical value of k is 1/8.
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