Question

Find each value without using a calculator $$ \sin \left[\cos ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{5}{13}\right)\right] $$

Short answer

The value is \(\frac{56}{65}\).

Step-by-step solution

Recall the Inverse Cosine Identity

To solve the problem, we first need to understand what \(\cos^{-1}(x)\) represents. The expression \(\cos^{-1}(x)\) is the angle whose cosine is \(x\). Thus, for \(\cos^{-1}\left(\frac{3}{5}\right)\), the cosine of this angle equals \(\frac{3}{5}\). Similarly, for \(\cos^{-1}\left(\frac{5}{13}\right)\), the cosine of this angle equals \(\frac{5}{13}\).

Identify the Angles

We will label these angles as follows:Let \(\theta_1 = \cos^{-1}\left(\frac{3}{5}\right)\) and \(\theta_2 = \cos^{-1}\left(\frac{5}{13}\right)\). This means that \(\cos(\theta_1) = \frac{3}{5}\) and \(\cos(\theta_2) = \frac{5}{13}\).

Use the Pythagorean Theorem to Find Sine Values

To find \(\sin(\theta_1)\) and \(\sin(\theta_2)\), we can use the identity \(\sin\theta = \sqrt{1 - \cos^2\theta}\).For \(\theta_1\):\[\sin(\theta_1) = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}\]For \(\theta_2\):\[\sin(\theta_2) = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13}\]

Apply the Sine Addition Formula

The expression requires us to find \(\sin(\theta_1 + \theta_2)\). We use the sine addition formula:\[\sin(\theta_1 + \theta_2) = \sin(\theta_1) \cos(\theta_2) + \cos(\theta_1) \sin(\theta_2)\]Substitute the known values:\(\sin(\theta_1) = \frac{4}{5}, \cos(\theta_1) = \frac{3}{5}, \sin(\theta_2) = \frac{12}{13}, \cos(\theta_2) = \frac{5}{13}\).\[\sin(\theta_1 + \theta_2) = \left(\frac{4}{5}\right)\left(\frac{5}{13}\right) + \left(\frac{3}{5}\right)\left(\frac{12}{13}\right)\]

Calculate the Final Expression

Compute the expression from the previous step:\[\sin(\theta_1 + \theta_2) = \frac{4 \times 5}{65} + \frac{3 \times 12}{65} = \frac{20}{65} + \frac{36}{65}\]Combine the fractions:\[\sin(\theta_1 + \theta_2) = \frac{20 + 36}{65} = \frac{56}{65}\]

Conclusion

The value of the expression \(\sin \left[\cos ^{-1}\left(\frac{3}{5}\right) + \cos^{-1}\left(\frac{5}{13}\right)\right]\) is \(\frac{56}{65}\).

Key concepts

Inverse Cosine

The inverse cosine function, denoted as \(\cos^{-1}(x)\), is used to find the angle whose cosine is \(x\). The function outputs an angle, commonly in the range from 0 to \(\pi\) radians (0 to 180 degrees), as cosine is positive in these regions. For example, when we have \(\cos^{-1}\left(\frac{3}{5}\right)\), it returns an angle \(\theta_1\) where the cosine of \(\theta_1\) equals \(\frac{3}{5}\). To visualize this, imagine a right triangle where the adjacent side over the hypotenuse equals \(\frac{3}{5}\).
Inverse trigonometric functions are crucial when you want to move from ratio to angle. They let you work backward from a given ratio to determine the actual angle, much like flipping the process of a regular cosine calculation.

• Real-world applications include calculating angles in physics problems and engineering designs.
• It's important to note that the output angle is always in the principal range, which keeps interpretations consistent.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in trigonometry that enables us to find missing side lengths in right triangles. The theorem states: \[ a^2 + b^2 = c^2 \] where \(a\) and \(b\) are the legs of the triangle and \(c\) is the hypotenuse. In trigonometry, this theorem helps us find sine or cosine values when one of them is known. For an angle \(\theta\), if we know \(\cos(\theta)\), we can find \(\sin(\theta)\) using the identity: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] This identity is derived directly from the Pythagorean theorem by considering a right triangle with a hypotenuse of 1.
For instance, given \(\cos(\theta_1) = \frac{3}{5}\), the sine can be calculated as \(\sqrt{1 - (\frac{3}{5})^2} = \frac{4}{5}\). This process helps in determining unknown trigonometric ratios when one is provided.

• The theorem is essential for solving triangles and finding trigonometric functions.
• It's used extensively in all fields of science and engineering where geometry comes into play.

Sine Addition Formula

The sine addition formula is a key identity in trigonometry used to find the sine of the sum of two angles. It states: \[ \sin(\theta_1 + \theta_2) = \sin(\theta_1)\cos(\theta_2) + \cos(\theta_1)\sin(\theta_2) \] This formula allows us to calculate the sine of combined angles using the sines and cosines of the individual angles. For example, with \(\theta_1 = \cos^{-1}\left(\frac{3}{5}\right)\) and \(\theta_2 = \cos^{-1}\left(\frac{5}{13}\right)\), we use known sine and cosine values for each angle to find \(\sin(\theta_1 + \theta_2)\): \[ \sin(\theta_1 + \theta_2) = \left(\frac{4}{5}\right)\left(\frac{5}{13}\right) + \left(\frac{3}{5}\right)\left(\frac{12}{13}\right) \] This concept simplifies complex angle calculations and is very useful in multi-angle problems or functions modeling.

• Helps in decomposing complex signals in electrical engineering and physics.
• Important in calculating rotational dynamics and oscillations.

Angles in Radians

Radians provide a natural way of expressing angles in mathematics, often preferred over degrees for calculus and trigonometry. One radian is defined as the angle created when the arc length is equal to the radius of a circle. In terms of degrees, \[ 1 \text{ radian} \approx 57.2958 \text{ degrees} \] For any circle, a complete revolution of 360 degrees equals \(2\pi\) radians. Converting angles to radians is straightforward:

• To convert degrees to radians, use \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).
• To convert radians to degrees, use \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \).

Understanding radians is key when solving problems involving trigonometric functions since they naturally fit into the periodic nature of functions such as sine and cosine. They also simplify derivatives and integrals in calculus, making them an essential part of mathematical analysis.