Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store
seo-qna
SearchIcon
banner

Find the value of sin315cos315+sin420cos330

Answer
VerifiedVerified
489k+ views
like imagedislike image
Hint: We solve this problem first by converting all the angles given to less than 90 because we have the trigonometric table of standard values for the angles less than 90
We convert each angle as the sum or difference of 360 that we convert each angle as 360+θ or 360θ so that we have conversions of trigonometric ratios as
sin(360+θ)=sinθ
sin(360θ)=sinθ
cos(360+θ)=cosθ
cos(360θ)=cosθ
By using the above formulas we reduce the given angles into angles less than 90 to find the required value easily.

Complete step by step answer:
We are asked to find the value of sin315cos315+sin420cos330
Let us assume that the required value as
A=sin315cos315+sin420cos330......equation(i)
Now, let us convert each angle in the above equation into 360+θ or 360θ
Now, by converting the first angle that is 315 we get
315=36045
Now, by converting the next angle that is 420 we get
420=360+60
Now, by converting the next angle that is 330 we get
330=36030
Now, by substituting the required angles in the equation (i) we get
A=sin(36045)cos(36045)+sin(360+60)cos(36030)
We know that the conversions of trigonometric ratios that are
sin(360+θ)=sinθ
sin(360θ)=sinθ
cos(360+θ)=cosθ
cos(360θ)=cosθ
Now, by using these conversions in the above equation we get
A=(sin45)(cos45)+(sin60)(cos30)A=sin45cos45sin60cos30
We know that from the standard trigonometric table the values of trigonometric ratios for some angles as
sin45=cos45=12sin60=cos30=32
Now, by substituting these values in above equation we get
A=(12)(12)(32)(32)A=1234A=14

Therefore we can conclude that the value of given expression as
sin315cos315+sin420cos330=14


Note: Students may do mistake in converting the given angles into angles less than 90
Here, we can see that the given angles are all near to 360 so that we can easily convert the given angles in the form 360+θ or 360θ to get the angles less than 90
But we have so many ways of converting the given angles into angles less than 90
We can convert in the form 450+θ or 450θ
But in this process the formulas of conversions will change, that is sine gets converted to cosine and cosine gets converted to sine and also the sign changes.
But even though we will not get the all angles less than 90 because
315=450135
Here, we again need to convert the angle 135 in the form of 90+θ which increases the solution length.
So, we need to check for nearest value where we get the given angles as the angles less than 90