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Question: If \(\sin \theta = \cos \theta \), then \(\theta = ?\) A.\({45^0}\) B.\({90^0}\) C.\({0^0}\) ...

If sin⁡θ=cos⁡θ\sin \theta = \cos \theta , then θ=?\theta = ?
A.450{45^0}
B.900{90^0}
C.00{0^0}
D.300{30^0}

Explanation

Solution

Hint : The trigonometric function is the function that relates the ratio of the length of two sides with the angles of the right-angled triangle widely used in navigation, oceanography, the theory of periodic functions, and projectiles. Commonly used trigonometric functions are the sine, the cosine, and the tangent, whereas the cosecant, the secant, the cotangent are their reciprocal, respectively.
In this question, we need to determine the angle θ\theta such that sin⁡θ=cos⁡θ\sin \theta = \cos \theta . For this we will use a general trigonometric identity which is given as cos⁡θ=sin⁡(π2−θ)\cos \theta = \sin \left( {\dfrac{\pi }{2} - \theta } \right).

Complete step-by-step answer :
The given equation in the question is sin⁡θ=cos⁡θ−−−−(i)\sin \theta = \cos \theta - - - - (i).
Using cos⁡θ=sin⁡(π2−θ)\cos \theta = \sin \left( {\dfrac{\pi }{2} - \theta } \right) in the equation (i), we get
sin⁡θ=cos⁡θ sin⁡θ=sin⁡(π2−θ)−−−−(ii)  \sin \theta = \cos \theta \\\ \sin \theta = \sin \left( {\dfrac{\pi }{2} - \theta } \right) - - - - (ii) \\\
If an equation has defined trigonometric function to both sides of an equation, then it can be eliminated leaving behind only the corresponding angles.
Here, in equation (ii) we can see that sin trigonometric function is common to both the sides of the equation so,
θ=(π2−θ)−−−−(iii)\theta = \left( {\dfrac{\pi }{2} - \theta } \right) - - - - (iii)
Now, solving equation (iii) for the value of θ\theta as
θ=(π2−θ) θ+θ=π2 2θ=π2 θ=π4  \theta = \left( {\dfrac{\pi }{2} - \theta } \right) \\\ \theta + \theta = \dfrac{\pi }{2} \\\ 2\theta = \dfrac{\pi }{2} \\\ \theta = \dfrac{\pi }{4} \\\
Now, substitute the value of θ\theta as 180 degrees for converting the result obtained in radians to degrees as:
θ=π4 =18004 =450  \theta = \dfrac{\pi }{4} \\\ = \dfrac{{{{180}^0}}}{4} \\\ = {45^0} \\\
Hence, if sin⁡θ=cos⁡θ\sin \theta = \cos \theta then, the value of θ\theta is 450{45^0}
So, the correct answer is “Option A”.

Note : Alternatively, following the defined table for the trigonometric terms.

Functionsin⁡θ\sin \theta cos⁡θ\cos \theta tan⁡θ\tan \theta
0010
300{30^0}12\dfrac{1}{2}32\dfrac{{\sqrt 3 }}{2}13\dfrac{1}{{\sqrt 3 }}
450{45^0}12\dfrac{1}{{\sqrt 2 }}12\dfrac{1}{{\sqrt 2 }}1
600{60^0}32\dfrac{{\sqrt 3 }}{2}12\dfrac{1}{2}3\sqrt 3
900{90^0}10Indeterminate

We can see that the value of sin and cosine terms are equal only at θ=π4=450\theta = \dfrac{\pi }{4} = {45^0}.