Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-36: Verify the reduction formula. $$ \cos \left(x+\frac{\pi}{2}\right)=-\sin x $$

Step-by-Step Solution

Answer

The reduction formula \( \cos(x + \frac{\pi}{2}) = - \sin x \) is verified true.

Step 1: Understand the Function

Identify the function that needs to be transformed. In this case, we have \( \cos(x + \frac{\pi}{2}) \). This means we are looking at the cosine of an angle that is \( \frac{\pi}{2} \) radians added to \( x \).

Step 2: Use the Cosine Angle Sum Formula

Recall the cosine angle sum identity, which is: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). Substitute \( a = x \) and \( b = \frac{\pi}{2} \). This gives us: \( \cos(x + \frac{\pi}{2}) = \cos x \cos \frac{\pi}{2} - \sin x \sin \frac{\pi}{2} \).

Step 3: Substitute Known Values

Apply the known values: \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \). Substitute these into the formula: \( \cos(x + \frac{\pi}{2}) = \cos x \cdot 0 - \sin x \cdot 1 \).

Step 4: Simplify the Expression

Simplify the expression obtained by carrying out the multiplication: \( \cos x \cdot 0 = 0 \) and \( - \sin x \cdot 1 = - \sin x \). Thus, the expression simplifies to: \( \cos(x + \frac{\pi}{2}) = - \sin x \).

Step 5: Conclusion

From the simplification, we can see that the reduction formula \( \cos(x + \frac{\pi}{2}) = - \sin x \) holds true, as the given identity matches the derived result.

Key Concepts

Cosine Angle Sum Formula

The cosine angle sum formula is a cornerstone in trigonometry. It allows us to find the cosine of the sum of two angles. The formula itself is expressed as:

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

This identity is extremely useful because it breaks down the cosine of a sum into simpler components. In order to apply this formula, all that's needed are the trigonometric values (cosine and sine) of the individual angles, \( a \) and \( b \).
For example, in our exercise, we substitute \( a = x \) and \( b = \frac{\pi}{2} \). This results in the expression:

• \( \cos(x + \frac{\pi}{2}) = \cos x \cos \frac{\pi}{2} - \sin x \sin \frac{\pi}{2} \)

We use the known values of \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \). By substituting these into our expression, we simplify it effectively. This process demonstrates how the cosine angle sum formula can make complex calculations more manageable.

Reduction Formulas

Reduction formulas are incredibly helpful in trigonometry as they link trigonometric functions involving sums or differences of angles to simpler or more familiar expressions. These equations transform complex trigonometric expressions by reducing them to functions of smaller angles. In the context of the original problem, the reduction formula we need is:

• \( \cos(x + \frac{\pi}{2}) = - \sin x \)

This particular formula comes in handy because it shows how the cosine function behaves when the angle is displaced by \( \frac{\pi}{2} \). To verify this reduction, we applied the cosine angle sum formula and showed how it simplifies directly to the known value.
By understanding reduction formulas, students can easily relate different trigonometric identities and find simplifications that are not immediately obvious. It's a powerful tool when dealing with trigonometric problems involving complicated angles, perfectly aiding in both complex problem-solving and straightforward calculations.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly in the study of triangles and oscillations. The primary trigonometric functions are sine, cosine, and tangent. Each of them describes a unique aspect of the unit circle:

• Sine (\( \sin \)): Measures the vertical component or height when projecting the angle onto the unit circle.
• Cosine (\( \cos \)): Measures the horizontal component or length when projecting it onto the unit circle.
• Tangent (\( \tan \)): Represents the ratio of sine to cosine, \( \tan x = \frac{\sin x}{\cos x} \).

These functions provide relationships among the angles and sides of right triangles. They extend beyond triangles to model periodic phenomena such as sound waves, light, and tides.
Our task with \( \cos(x + \frac{\pi}{2}) \) specifically deals with cosine and its transformation properties using the angle sum and reduction formulas. With a solid grasp of these foundational functions, students can venture into understanding the nature of periodicity, symmetry, and amplitude in various scientific fields.