CBSE · Class 10 · Mathematics · Introduction to TrigonometryIf $\sin (A - B) = \frac{1}{2}$ and $\cos (A + B) = \frac{1}{2}$, where $(A + B)$ is an acute angle and $A > B$, find the values of the angles $A$ and $B$. Explain the concept of complementary angles in trigonometry.
Concept of Complementary Angles in Trigonometry\nIn geometry, two angles are called complementary if their sum is equal to $90^\circ$. In the context of a right-angled triangle, since one angle is always $90^\circ$, the sum of the remaining two acute angles is also $90^\circ$. Therefore, if one acute angle is $\theta$, the other acute angle is $(90^\circ - \theta)$.
\nTrigonometric ratios of complementary angles establish standard relationships between the primary ratios of an angle and its complement:
- $\sin(90^\circ - \theta) = \cos \theta$
- $\cos(90^\circ - \theta) = \sin \theta$
- $\tan(90^\circ - \theta) = \cot \theta$
- $\cot(90^\circ - \theta) = \tan \theta$
- $\sec(90^\circ - \theta) = \csc \theta$
- $\csc(90^\circ - \theta) = \sec \theta$ \nThese relationships are extremely useful in simplifying complex trigonometric expressions and solving equations where direct angle values are not readily apparent.
Step-by-Step Solution for $A$ and $B$
Given Equations:
- $\sin (A - B) = \frac{1}{2}$
- $\cos (A + B) = \frac{1}{2}$
Step 1: Find the angle for the first equation.\nWe know from standard trigonometric table values that: $$\sin 30^\circ = \frac{1}{2}$$\nComparing this with $\sin (A - B) = \frac{1}{2}$: $$A - B = 30^\circ \quad \text{--- (Equation I)}$$
Step 2: Find the angle for the second equation.\nWe know from standard trigonometric table values that: $$\cos 60^\circ = \frac{1}{2}$$\nComparing this with $\cos (A + B) = \frac{1}{2}$: $$A + B = 60^\circ \quad \text{--- (Equation II)}$$
Step 3: Solve the system of linear equations.\nAdd Equation I and Equation II: $$(A - B) + (A + B) = 30^\circ + 60^\circ$$ $$2A = 90^\circ$$ $$A = \frac{90^\circ}{2} = 45^\circ$$
Step 4: Find the value of $B$.\nSubstitute the value of $A = 45^\circ$ into Equation II: $$45^\circ + B = 60^\circ$$ $$B = 60^\circ - 45^\circ$$ $$B = 15^\circ$$
Step 5: Verify the conditions.
- $A = 45^\circ$ and $B = 15^\circ$, so $A > B$ (Satisfied)
- $A + B = 45^\circ + 15^\circ = 60^\circ$, which is an acute angle (Satisfied)
Final Answer:\nThe value of angle $A$ is $45^\circ$ and the value of angle $B$ is $15^\circ$.