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Chapter 5: Problem 57

Find the exact value of each expression. Do not use a calculator. $$\cos 157.5^{\circ} \sin 22.5^{\circ}$$

Short Answer

Expert verified
The exact value of the expression is \(-\sqrt{2}/4.\)

Step by step solution

01

Write down the expression

The expression is \(\cos 157.5^{\circ} \sin 22.5^{\circ}\).
02

Apply the product-to-sum formula

The formula to use is \(\cos a \sin b = 1/2 [\sin(a+b) - \sin(a-b)]\). With \(a = 157.5^{\circ}\) and \(b = 22.5^{\circ}\), the expression becomes \(1/2 [\sin(157.5^{\circ}+22.5^{\circ}) - \sin(157.5^{\circ}-22.5^{\circ})]\).
03

Simplify

Simplify the angles inside the sin functions to \(\sin 180^{\circ}\) and \(\sin 135^{\circ}\). So, the expression becomes \(1/2 [\sin(180^{\circ}) - \sin(135^{\circ})]\).
04

Compute the sine values

Using knowledge of the unit circle, it is known that \(\sin 180^{\circ} = 0\) and \(\sin 135^{\circ} = \sqrt{2}/2\). The original expression simplifies to \(1/2 [0 - \sqrt{2}/2]\).
05

Compute the final expression

Finally, computation simplifies the expression to \(-\sqrt{2}/4.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product-to-Sum Formulas
Trigonometry has some fantastic identities, and one of these is the product-to-sum formula. It allows us to transform a product of trigonometric functions into a sum or difference. This can simplify calculations that would otherwise be tricky. For this exercise:
  • The formula used is: \[\cos a \sin b = \frac{1}{2} [\sin(a+b) - \sin(a-b)]\]This formula directly applies to the expression \(\cos 157.5^{\circ} \sin 22.5^{\circ}\).
  • By converting a product into a sum or difference, we can solve trigonometric expressions with ease.
  • It's particularly useful when working with angles that are not standard on the unit circle.
Unit Circle
Understanding the unit circle is foundational in trigonometry. It's a circle with a radius of one, centered at the origin of a coordinate system. On this circle:
  • Every angle in degrees or radians has corresponding coordinates, \((\cos \theta, \sin \theta)\).
  • At key angles like \(0^{\circ}, 90^{\circ}, 180^{\circ}, \) and \(270^{\circ}\), the sine and cosine values become very straightforward.
  • This helps us easily find trigonometric values for angles like \(135^{\circ}\) and \(180^{\circ}\), which were used in this exercise.
Here, knowing the values for \(\sin 180^{\circ} = 0\) and \(\sin 135^{\circ} = \sqrt{2}/2\) plays a crucial role in simplifying our original expression.
Angle Simplification
Manipulating angles into simpler calculations can dramatically ease solving trigonometric expressions. In this case:
  • Angles \(157.5^{\circ}\) and \(22.5^{\circ}\) are added and subtracted, simplifying to \(180^{\circ}\) and \(135^{\circ}\) respectively, before applying the sine function.
  • This technique is highly effective as \(180^{\circ}\) and \(135^{\circ}\) are well-known angles on the unit circle with standard sine values.
  • Angle simplification is key to breaking down complex problems into manageable steps, enhancing both speed and accuracy.
By understanding angle simplification, students can approach similar problems with more confidence.
Sine Values
Sine values are a fundamental aspect of trigonometry. They're linked directly to the y-coordinates on the unit circle and provide crucial information:
  • The sine of an angle \(\theta\) in a right triangle is the ratio of the opposite side to the hypotenuse.
  • On the unit circle, the sine value correlates with the vertical distance from the x-axis.
  • Key angles like \(180^{\circ}\) produce values such as \(\sin 180^{\circ} = 0\).
  • Likewise, \(\sin 135^{\circ} = \sqrt{2}/2\) due to its position in the second quadrant, where sine values are positive.
In this exercise, knowing these sine values helped ease the calculation process. The end result, \(-\sqrt{2}/4\), comes from these precise unit circle values, ensuring accuracy without tools like calculators.

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