Using the sum formula for cosine, express $$ \cos(a + b) $$ in terms of $$ \cos a, \cos b, \sin a, ext{ and } \sin b $$.

Question

Using the sum formula for cosine, express $$ \cos(a + b) $$ in terms of $$ \cos a, \cos b, \sin a, 	ext{ and } \sin b $$.

UpStudy AI · Accepted Answer

The sum formula for cosine allows you to express $$ \cos(a + b) $$ in terms of the cosines and sines of the individual angles $$ a $$ and $$ b $$. The formula is derived from the addition of angles in trigonometry and is given by:

$$
\cos(a + b) = \cos a \cos b - \sin a \sin b
$$

**Explanation:**

1. **Cosine of a Sum:**
   The cosine of the sum of two angles $$ a $$ and $$ b $$ can be expanded using the sum formula:
   $$
   \cos(a + b) = \cos a \cos b - \sin a \sin b
   $$

2. **Components:**
   - $$ \cos a \cos b $$: This term represents the product of the cosines of the individual angles.
   - $$ \sin a \sin b $$: This term represents the product of the sines of the individual angles.
   - The minus sign between the two terms indicates that the sine product is subtracted from the cosine product.

**Example Usage:**

If you want to calculate $$ \cos(30^\circ + 45^\circ) $$, you can apply the formula as follows:

$$
\cos(75^\circ) = \cos 30^\circ \cos 45^\circ - \sin 30^\circ \sin 45^\circ
$$

Using known values:
$$
\cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 30^\circ = \frac{1}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}
$$

Plugging these in:
$$
\cos(75^\circ) = \left( \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} \right) - \left( \frac{1}{2} \times \frac{\sqrt{2}}{2} \right) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}
$$

This formula is fundamental in trigonometry and is widely used in various applications, including solving triangles, oscillatory motion, and wave analysis.

**Summary Formula:**
$$
\boxed{\cos(a + b) = \cos a\, \cos b\; -\; \sin a\, \sin b}
$$
$$ \cos(a + b) = \cos a \cos b - \sin a \sin b $$

Using the sum formula for cosine, express in terms of .

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Trigonometry Questions