Problem 5 Show that $$ \begin{array}{l... [FREE SOLUTION]

Chapter 8: Problem 5

Show that $$ \begin{array}{l} \sin \left(\frac{\pi}{4}\right)=\cos \left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}} \\ \sin \left(\frac{\pi}{6}\right)=\cos \left(\frac{\pi}{3}\right)=\frac{1}{2} \end{array} $$ and $$ \sin \left(\frac{\pi}{3}\right)=\cos \left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2} $$

Short Answer

Expert verified

The identities are verified using right triangle properties.

Step by step solution

Use Right Triangle Properties

The angles $ \frac{\pi}{4}, \frac{\pi}{3}, \text{ and } \frac{\pi}{6} $ correspond to 45掳, 60掳, and 30掳, respectively. Recognize these as standard angles often found in right triangles with angles of 45-45-90 and 30-60-90.

Determine Ratios for 45-45-90 Triangle

In a 45-45-90 triangle, the two legs are equal and the hypotenuse is $ \sqrt{2} $ times the length of a leg. Therefore, the sine and cosine for 45掳 (or $ \frac{\pi}{4} $) are $ \frac{1}{\sqrt{2}} $.

Determine Ratios for 30-60-90 Triangle: 30掳 Angle

In a 30-60-90 triangle, the side opposite the 30掳 angle ( $ \frac{\pi}{6} $ ) is half the hypotenuse, giving $ \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} $, and the adjacent side is half the hypotenuse times $ \sqrt{3} $, so $ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} $.

Determine Ratios for 30-60-90 Triangle: 60掳 Angle

Continuing with the 30-60-90 triangle, the side opposite the 60掳 angle ( $ \frac{\pi}{3} $ ) is half the hypotenuse times $ \sqrt{3} $, which shows $ \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $, and the adjacent side corresponds to $ \cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} $.

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Key Concepts

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

Right Triangle Properties

Right triangles are a key concept in trigonometry, characterized by the presence of a 90-degree angle. These triangles have specific properties that make calculating the sine and cosine of standard angles straightforward. Right triangles are often classified by their angle combinations, such as 45-45-90 and 30-60-90 triangles. Each configuration helps us derive particular trigonometric ratios.

In a 45-45-90 triangle:

Both non-right angles are 45掳.
The lengths of the legs are equal.
The hypotenuse is $ \sqrt{2} $ times the length of each leg.

In a 30-60-90 triangle:

The angles are 30掳, 60掳, and 90掳.
The hypotenuse is twice the length of the shortest side (opposite the 30掳 angle).
The longer leg is $ \sqrt{3} $ times the shortest side (between the 60掳 and 90掳 angles).

Understanding these properties simplifies the process of determining trigonometric ratios for these standard angles.

Standard Angles

Standard angles refer to specific commonly used angles where trigonometric ratios are often memorized due to their frequent occurrence in problems. These include angles like 0掳, 30掳, 45掳, 60掳, and 90掳.

For the following standard angles, we can understand their link to common right triangle configurations to derive various trigonometric functions:

At 45掳 $ (\frac{\pi}{4}) $, the triangle is an isosceles right triangle (45-45-90), where the sine and cosine both equal $ \rac{1}{\sqrt{2} } $.
At 30掳 $ (\frac{\pi}{6}) $, found in the 30-60-90 triangle, the opposite side to 30掳 is half the hypotenuse, giving us $ \sin(\frac{\pi}{6}) = \frac{1}{2} $.
At 60掳 $ (\frac{\pi}{3}) $, also in a 30-60-90 triangle, the side opposing the 60掳 angle becomes $ \frac{\sqrt{3}}{2} $ times the hypotenuse, providing $ \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} $.

These angles and their respective triangles are invaluable for calculating exact values without a calculator.

Triangle Ratios

Trigonometric ratios are measures that relate the angles to the sides of a right triangle. The primary ratios are sine (\sin ), cosine (\cos), and tangent (\tan). Let's focus on sine and cosine as they directly pertain to the problem:

For any angle in a right triangle:

The sine of an angle is the ratio of the length of the opposite side to the hypotenuse. Formulaically, $ \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} $.
The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. $ \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} $.

Exploring specific triangles:

In a 45-45-90 triangle, since the legs are the same, both sine and cosine of 45掳 yield $ \frac{1}{\sqrt{2}} $.
In a 30-60-90 triangle, $ \sin(30掳) = \frac{1}{2} $ because the side opposite 30掳 is half the hypotenuse, and $ \sin(60掳) $ becomes $ \frac{\sqrt{3}}{2} $ as the opposite side is $ \sqrt{3} $ times longer.
Cosine values for 60掳 and 30掳 are switched vice versa to the sine values because of their positional changes as the triangle rotates.

These ratios are crucial as they provide easy, memorized values that can quickly solve problems involving right triangles without needing extensive calculations.

91影视

Show that $$ \begin{array}{l} \sin \left(\frac{\pi}{4}\right)=\cos \left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}} \\ \sin \left(\frac{\pi}{6}\right)=\cos \left(\frac{\pi}{3}\right)=\frac{1}{2} \end{array} $$ and $$ \sin \left(\frac{\pi}{3}\right)=\cos \left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2} $$

Short Answer

Step by step solution

Use Right Triangle Properties

Determine Ratios for 45-45-90 Triangle

Determine Ratios for 30-60-90 Triangle: 30掳 Angle

Determine Ratios for 30-60-90 Triangle: 60掳 Angle

Key Concepts

Right Triangle Properties

Standard Angles

Triangle Ratios

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