wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

The value of sinθ+cosθ will be greatest when


A

θ=30°

No worries! We‘ve got your back. Try BYJU‘S free classes today!
B

θ=45°

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C

θ=60°

No worries! We‘ve got your back. Try BYJU‘S free classes today!
D

θ=90°

No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

The correct option is B

θ=45°


Explanation for the correct option:

Step 1. Find the critical point

The given trigonometric expression: sinθ+cosθ.

Let us assume that, fθ=sinθ+cosθ.

Differentiate both sides of the equation with respect to θ.

ddθfθ=ddθsinθ+cosθf'θ=cosθ-sinθ

For critical point f'θ=0

cosθ-sinθ=0sinθ=cosθtanθ=1θ=π4,5π4

Step 2. Find the point of maxima

For maxima f''θ<0

Differentiate both sides of the equation with respect to θ.

ddθf'θ=ddθcosθ-sinθf''θ=-sinθ-cosθ

Now, f''π4=-sinπ4-cosπ4.

f''π4=-12-12f''π4=-22f''π4=-2

Thus, f''π4<0.

Now, f''5π4=-sin5π4-cos5π4.

f''5π4=12+12f''5π4=22f''5π4=2

Thus, f''5π4>0

Thus, fθ has a maximum value at θ=π4.

Therefore, the value of sinθ+cosθ will be greatest when θ=π4=45°.

Hence, the correct option is (B).


flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Second Derivative Test for Local Maximum
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon