Question

a) Determine the positive value of \(\sin \left(\cos ^{-1} 0.6\right) .\) Use your knowledge of the unit circle to explain why the answer is a rational number. b) Without calculating, what is the positive value of \(\cos \left(\sin ^{-1} \mathbf{0 . 6}\right) ?\) Explain.

Short answer

a) \(\sin(\cos^{-1}(0.6)) = 0.8\). b) \(\cos(\sin^{-1}(0.6)) = 0.8\).

Step-by-step solution

Understand the Problem

We are asked to find \(\sin (\cos^{-1} 0.6)\). This involves understanding the inverse trigonometric functions and how to find the sine of an angle whose cosine is 0.6.

Define \(\theta\)

Let \(\theta = \cos^{-1}(0.6)\). This means \(\cos(\theta) = 0.6\). We need to find \(\sin(\theta)\).

Use the Pythagorean Identity

Recall the Pythagorean identity: \[\cos^2(\theta) + \sin^2(\theta) = 1\] Since \(\cos(\theta) = 0.6\), we can substitute: \[0.6^2 + \sin^2(\theta) = 1\]

Solve for \(\sin(\theta)\)

Simplify the equation: \[0.36 + \sin^2(\theta) = 1\] Subtract 0.36 from both sides: \[\sin^2(\theta) = 0.64\] Take the positive square root of both sides: \[\sin(\theta) = \sqrt{0.64} = 0.8\]

Conclusion for Part (a)

Since \(\sin(\theta) = 0.8\), the positive value of \(\sin(\cos^{-1}(0.6))\) is 0.8. Using the unit circle, any angle that has a cosine of 0.6 will have a sine of 0.8, a rational number.

Understand Part (b)

Now, we need to find \(\cos(\sin^{-1}(0.6))\). This means we need the cosine of an angle whose sine is 0.6.

Define \(\alpha\)

Let \(\alpha = \sin^{-1}(0.6)\). This means \(\sin(\alpha) = 0.6\). We now need to find \(\cos(\alpha)\).

Use the Pythagorean Identity Again

Recall the identity: \[\sin^2(\alpha) + \cos^2(\alpha) = 1\] Since \(\sin(\alpha) = 0.6\), we substitute: \[0.6^2 + \cos^2(\alpha) = 1\]

Solve for \(\cos(\alpha)\)

Simplify the equation: \[0.36 + \cos^2(\alpha) = 1\] Subtract 0.36 from both sides: \[\cos^2(\alpha) = 0.64\] Take the positive square root: \[\cos(\alpha) = \sqrt{0.64} = 0.8\]

Conclusion for Part (b)

Therefore, the positive value of \(\cos(\sin^{-1}(0.6))\) is also 0.8. No calculation was needed because the problem is symmetric in this context.

Key concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It states that for any angle \( \theta \), the square of the sine of the angle plus the square of the cosine of the angle equals one. In mathematical terms, it is expressed as: \[ \cos^2(\theta) + \sin^2(\theta) = 1 \].
This identity is derived from the Pythagorean Theorem and the definition of sine and cosine on the unit circle.
For instance, if you know the cosine of an angle is 0.6, you can find the sine using the identity. By substituting \( \cos(\theta) = 0.6 \) into the equation, you'll get: \[ 0.6^2 + \sin^2(\theta) = 1 \].
Simplify to find \( \sin^2(\theta) \) and then take the positive square root.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It plays a key role in trigonometry, especially when dealing with sine and cosine functions. Each point on the unit circle corresponds to an angle and has coordinates \( (\cos(\theta), \sin(\theta)) \). This setup helps to visualize why \( \cos^2(\theta) + \sin^2(\theta) = 1 \) — because it represents the distance from the origin to a point on the circle, which must always be 1 according to the Pythagorean Theorem.
Understanding the unit circle allows you to easily see relationships between angles, sine, and cosine. For example, if the cosine of an angle is 0.6, you can find the corresponding sine using the unit circle and the Pythagorean Identity, as shown above.

Sine and Cosine Relationships

Sine and cosine are deeply interconnected through various identities and relationships. One of the most fundamental relationships is encapsulated in the Pythagorean Identity, but there are several other essential connections:

• Complementary Angles: \( \sin(\theta) = \cos(90^\circ - \theta) \)
• Co-function Relationships: \( \cos(\theta) = \sin(90^\circ - \theta) \)
• Even-Odd Properties: \( \cos(-\theta) = \cos(\theta) \) and \( \sin(-\theta) = -\sin(\theta) \)

These relationships stem from how sine and cosine are defined on the unit circle and how they interact geometrically. Hence, understanding one function often gives you insights into the other, making it easier to solve trigonometric problems.
For example, if you know \( \cos(\theta) \) and need to find \( \sin(\theta) \), you can leverage these relationships alongside the Pythagorean Identity to find your answer.