Question
Question: The exact value of \[\cos \dfrac{{2\pi }}{{28}}\cos ec\dfrac{{3\pi }}{{28}} + \cos \dfrac{{6\pi }}{{...
The exact value of cos282πcosec283π+cos286πcosec289π+cos2818πcosec2827π is?
A.−21 B.21 C.1 D.0
Solution
Suitable uses of trigonometry identities is must here. Also apply complementary formulas of sin and cos values. Inverse ratios are also applicable here. Many sequential steps will be needed to reach the required result. Formula like sin(A+B) , cos(A+B) etc will be needed.
Complete step-by-step solution:
GIven cos282πcosec283π+cos286πcosec289π+cos2818πcosec2827π ……….…(1)
Let us assume that 28π=θ ……….…(2)
Then putting above value from equation (2), in equation (1), expression will be now,
cos2θcosec3θ+cos6θcosec9θ+cos18θcosec27θ ……...…(3)
Now, replace cosec terms by their reciprocal sine terms, as follows
sin3θcos2θ+sin9θcos6θ+sin27θcos18θ
Further, we get with simplification,
sin3θcos2θ+sin(14θ−5θ)cos6θ+sin(28θ−θ)cos(14θ+4θ) ………....(4)
As 28π=θ , then we get
2π=14θ ……...….(5)
Using the value from equations (5) and (2) in equation (4), we get
sin3θcos2θ+sin(2π−5θ)cos6θ+sin(π−θ)cos(2π+4θ)
Since we know that, sin(2π−A)=cosA and sin(π−A)=cosA and cos(2π+A)=−sinA .
So above expression becomes,
⇒sin3θcos2θ+cos5θcos6θ+sinθ−sin4θ
⇒sin5θcos5θsinθcos2θcos5θsinθ+cos6θsin3θsinθ−sin4θcos5θsin3θ …………..(6)
Now, we will simplify the numerator of above expression, as below,
cos2θcos5θsinθ+cos6θsin3θsinθ−sin4θcos5θsin3θ
⇒21[(2cos2θcos5θ)sinθ+cos6θ(2sin3θsinθ)] ⇒21[(cos7θ+cos3θ)sinθ+cos6θ(−cos4θ+cos2θ)−cos5θ(−cos2θ+cosθ)] ⇒41[2sinθcos7θ+2sinθcos3θ−2cos6θcos4θ+2cos6θcos2θ+2cos5θcos7θ−2cos5θcosθ]
Now, we do further simplification, then as below,
⇒41[sin(4θ)−sin(2θ)+sin8θ−sin6θ+cos8θ+cos4θ−cos10θ−cos2θ−cos6θ−cos4θ+cos12θ+cos2θ]
Then
⇒41[sin(14θ−10θ)−sin(14θ−12θ)+sin(14θ−6θ)−sin(14θ−8θ)+cos8θ+cos4θ−cos10θ−cos2θ−cos6θ−cos4θ+cos12θ+cos2θ]since we have the value of 14θ as 2π from equation (5). So above expression will become,
⇒41[cos10θ−cos12θ+cos6θ−cos8θ+cos8θ−cos10θ−cos2θ−cos6θ+cos12θ+cos2θ] ⇒41×0 ⇒0
Thus with the help of above simplification we substitute this value in equation (6) , then we get
⇒sin5θcos5θsinθcos2θcos5θsinθ+cos6θsin3θsinθ−sin4θcos5θsin3θ=sin5θcos5θsinθ0 ⇒0
∴ The correct option is D.
Note: Trigonometry is the study of relationships between angles, lengths, and heights of triangles. Also, it shows the relationship between different parts of circles and other geometrical figures. Trigonometric identities are useful and hence its learning is very much required for solving the problems in a better way. There are many fields from science also, where these identities of trigonometry and formula of trigonometry are used.
One must know the difference between Trigonometric identities and Trigonometric Ratios. Trigonometric Identities are the formulas involving the trigonometric functions. Whereas, trigonometric Ratio is known for the relationship between the angles and the length of the side of the right triangle.