Trigonometric Expression Simplification
The problem requires finding the value of the expression $\sin(48^\circ + \theta) - \cos(42^\circ - \theta)$.
Applying Co-function Identity
We use the trigonometric co-function identity: $\cos(x) = \sin(90^\circ - x)$.
Applying this to the term $\cos(42^\circ - \theta)$, we set $x = 42^\circ - \theta$.
$ \cos(42^\circ - \theta) = \sin\left(90^\circ - (42^\circ - \theta)\right) $
Simplify the angle:
$ \cos(42^\circ - \theta) = \sin\left(90^\circ - 42^\circ + \theta\right) $
$ \cos(42^\circ - \theta) = \sin\left(48^\circ + \theta\right) $
Evaluating the Expression
Substitute the simplified term back into the original expression:
$ \sin(48^\circ + \theta) - \cos(42^\circ - \theta) = \sin(48^\circ + \theta) - \sin(48^\circ + \theta) $
Performing the subtraction:
$ \sin(48^\circ + \theta) - \sin(48^\circ + \theta) = 0 $
Result
The value of the expression $\sin(48^\circ + \theta) - \cos(42^\circ - \theta)$ is $0$. The correct option is D.
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What is the ratio of the greatest to the smallest value of 2 – 2 sin x – sin 2x, 0 ≤ x ≤ (π/2)?
If sinθ = \(\frac{4}{5}\) , Find the value of sin3θ
If x, y are acute angles, where 0 < x + y < 90° and sin(3x - 40°) = cos (3y + 40°), then the value of tan (x + y) is equal to
What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
If \(\sin \theta =\frac{3}{5}\) and \(\cos \theta =\frac{4}{5}\) , then the value of \(\frac{1+\tan \theta}{1-\cot \theta}\) is:
