Problem 100
Question
Find the exact value of each expression. Do not use a calculator. $$ \sin \left[\sin ^{-1} \frac{3}{5}-\cos ^{-1}\left(-\frac{4}{5}\right)\right] $$
Step-by-Step Solution
Verified Answer
The exact value of the given expression is -24/25.
1Step 1: Evaluate the Inverse sine and cosine
The inverse sine of \(\frac{3}{5}\) is an angle whose sine is \(\frac{3}{5}\). Similarly, the inverse cosine of \(-\frac{4}{5}\) is an angle whose cosine is \(-\frac{4}{5}\). Let's denote them as \(\theta\) and \(\phi\), respectively. Therefore, we have \(\sin \theta = \frac{3}{5}\) and \(\cos \phi = -\frac{4}{5}\).
2Step 2: Use the Pythagorean identity for sine and cosine
The Pythagorean identity for sine and cosine is \(\sin^2 x + \cos^2 x = 1\). By using this identity, we can find the values of \(\cos \theta \) and \(\sin \phi \). For \(\theta\), by substituting \(\sin \theta = \frac{3}{5}\) into the identity and solving for \(\cos \theta\), we get \(\cos \theta = \frac{4}{5}\). Similarly, for \(\phi\), by substituting \(\cos \phi = -\frac{4}{5}\) into the identity and solving for \(\sin \phi\), we get \(\sin \phi = \frac{3}{5}\).
3Step 3: Begin to simplify the expression
Now, let's substitute these values into the original problem. The problem becomes: \(\sin [ \theta - \phi ]\).
4Step 4: Use the difference of two angles identity
The difference of two angles identity for sine is \(\sin (a - b) = \sin a \cos b - \cos a \sin b\). Therefore, by substituting the values we got from Steps 1 and 2 into this identity, we get \(\sin [ \theta - \phi ] = \sin \theta \cos \phi - \cos \theta \sin \phi = \frac{3}{5} \cdot -\frac{4}{5} - \frac{4}{5} \cdot \frac{3}{5} = -\frac{24}{25}\).
Key Concepts
Inverse Trigonometric FunctionsPythagorean IdentityAngle Difference IdentitiesExact Values in Trigonometry
Inverse Trigonometric Functions
Inverse trigonometric functions are used to find angles from known trigonometric ratios. These functions include inverse sine (also written as arcsin), inverse cosine (arccos), and inverse tangent (arctan). For example, if you know that the sine of some angle is \(\frac{3}{5}\), the inverse sine function helps you find this specific angle.
This function basically reverses the sine operation. If \(\theta\) is the angle found, it would mean \(\sin^{-1}\left(\frac{3}{5}\right) = \theta\). Similarly, if the cosine of an angle is \(-\frac{4}{5}\), the inverse cosine function helps find this angle, denoted as \(\phi\), fulfilling the condition \(\cos^{-1}(-\frac{4}{5}) = \phi\).
Inverse functions are immensely useful in trigonometry, especially when trying to solve equations and finding angles that yield specific sine or cosine values.
This function basically reverses the sine operation. If \(\theta\) is the angle found, it would mean \(\sin^{-1}\left(\frac{3}{5}\right) = \theta\). Similarly, if the cosine of an angle is \(-\frac{4}{5}\), the inverse cosine function helps find this angle, denoted as \(\phi\), fulfilling the condition \(\cos^{-1}(-\frac{4}{5}) = \phi\).
Inverse functions are immensely useful in trigonometry, especially when trying to solve equations and finding angles that yield specific sine or cosine values.
Pythagorean Identity
The Pythagorean Identity is one of the fundamental identities in trigonometry, given as \(\sin^2 x + \cos^2 x = 1\). This identity simplifies the process of finding one trigonometric ratio when you know the other.
If you know \(\sin \theta = \frac{3}{5}\), for instance, you can find \(\cos \theta\) using the Pythagorean Identity: simply plug the known sine value into the equation and solve. You compute \(\cos \theta\) by rearranging it as \(\cos \theta = \sqrt{1 - \sin^2 \theta}\), resulting in \(\cos \theta = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5}\).
This identity also helps when converting a negative cosine value, such as \(-\frac{4}{5}\) for \(\phi\), into its corresponding sine value: \(\sin \phi = \frac{3}{5}\). It highlights the relationship between sine and cosine for any given angle.
If you know \(\sin \theta = \frac{3}{5}\), for instance, you can find \(\cos \theta\) using the Pythagorean Identity: simply plug the known sine value into the equation and solve. You compute \(\cos \theta\) by rearranging it as \(\cos \theta = \sqrt{1 - \sin^2 \theta}\), resulting in \(\cos \theta = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5}\).
This identity also helps when converting a negative cosine value, such as \(-\frac{4}{5}\) for \(\phi\), into its corresponding sine value: \(\sin \phi = \frac{3}{5}\). It highlights the relationship between sine and cosine for any given angle.
Angle Difference Identities
Angle difference identities are key when dealing with expressions like \(\sin (a - b)\). For sine specifically, the identity is \(\sin (a - b) = \sin a \cdot \cos b - \cos a \cdot \sin b\). These identities allow us to break down the calculation into finer, more manageable parts.
In the expression \(\sin [\theta - \phi]\), where you know \(\sin \theta\), \(\cos \theta\), \(\sin \phi\), and \(\cos \phi\), you can plug these values directly into the identity for simplified computation.
Here, \(\sin (\theta - \phi) = \left(\frac{3}{5}\right)\left(-\frac{4}{5}\right) - \left(\frac{4}{5}\right)\left(\frac{3}{5}\right) = -\frac{24}{25}\). The angle difference identities are a powerful tool for handling complex angle expressions without a calculator.
In the expression \(\sin [\theta - \phi]\), where you know \(\sin \theta\), \(\cos \theta\), \(\sin \phi\), and \(\cos \phi\), you can plug these values directly into the identity for simplified computation.
Here, \(\sin (\theta - \phi) = \left(\frac{3}{5}\right)\left(-\frac{4}{5}\right) - \left(\frac{4}{5}\right)\left(\frac{3}{5}\right) = -\frac{24}{25}\). The angle difference identities are a powerful tool for handling complex angle expressions without a calculator.
Exact Values in Trigonometry
Exact values in trigonometry refer to specific angle values for which the sine and cosine functions yield well-known numerical results, often derived from perfect triangles like the 45-45-90 or 30-60-90 triangles.
Using these values is critical for exercises requiring precision, such as this one where you're tasked to avoid calculator approximations. This methodology honors the geometric roots of trigonometric functions by relying on known ratios of sides in these triangles.
For example, the solution involving \(\sin^{-1}\left(\frac{3}{5}\right)\) and \(\cos^{-1}\left(-\frac{4}{5}\right)\) involves these precise calculations. Knowing trigonometric identities and how to use inverse functions are part of accessing these exact values and answering problems accurately.
Using these values is critical for exercises requiring precision, such as this one where you're tasked to avoid calculator approximations. This methodology honors the geometric roots of trigonometric functions by relying on known ratios of sides in these triangles.
For example, the solution involving \(\sin^{-1}\left(\frac{3}{5}\right)\) and \(\cos^{-1}\left(-\frac{4}{5}\right)\) involves these precise calculations. Knowing trigonometric identities and how to use inverse functions are part of accessing these exact values and answering problems accurately.
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