The problem presented is:

$\text{tan}\left( \arccos\left(\frac{5}{13}\right) + \arcsin\left(\frac{3}{5}\right)\right) = ?$

Let's break this down step by step.

Step 1: Understanding the values

• $\arccos\left(\frac{5}{13}\right)$ gives the angle whose cosine is $\frac{5}{13}$.
• $\arcsin\left(\frac{3}{5}\right)$ gives the angle whose sine is $\frac{3}{5}$.

Step 2: Use of Pythagoras for $\arccos\left(\frac{5}{13}\right)$

For $\arccos\left(\frac{5}{13}\right)$, we have:

• Adjacent side = 5
• Hypotenuse = 13

Using the Pythagorean theorem to find the opposite side: $\text{opposite} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ Thus, $\sin(\theta) = \frac{12}{13}$ for the angle $\theta = \arccos\left(\frac{5}{13}\right)$.

Step 3: For $\arcsin\left(\frac{3}{5}\right)$

For $\arcsin\left(\frac{3}{5}\right)$, we have:

• Opposite side = 3
• Hypotenuse = 5

Using the Pythagorean theorem to find the adjacent side: $\text{adjacent} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$ Thus, $\cos(\phi) = \frac{4}{5}$ for the angle $\phi = \arcsin\left(\frac{3}{5}\right)$.

Step 4: Using the tangent addition formula

We now want to compute: $\text{tan}(\theta + \phi)$ Using the identity for the tangent of a sum of two angles: $\text{tan}(\theta + \phi) = \frac{\text{tan}(\theta) + \text{tan}(\phi)}{1 - \text{tan}(\theta)\text{tan}(\phi)}$ We already know: $\text{tan}(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{12}{5}, \quad \text{tan}(\phi) = \frac{3}{4}$

Now, applying the tangent addition formula: $\text{tan}(\theta + \phi) = \frac{\frac{12}{5} + \frac{3}{4}}{1 - \left(\frac{12}{5} \cdot \frac{3}{4}\right)}$

Step 5: Simplifying the expression

Let's calculate the sum and product in the formula. I'll compute this.The value of $\tan\left(\arccos\left(\frac{5}{13}\right) + \arcsin\left(\frac{3}{5}\right)\right)$ is approximately $-3.9375$.

Would you like a detailed explanation of any specific step, or have any further questions?

Here are 5 related questions to deepen your understanding:

1. How is the Pythagorean theorem used to find the missing side in right-angled triangles?
2. What is the tangent addition formula, and when is it useful?
3. Why does $\arccos$ and $\arcsin$ yield angles whose trigonometric identities are useful in such problems?
4. How do we simplify complex trigonometric expressions like $\tan(\theta + \phi)$?
5. What are the domain and range limitations for inverse trigonometric functions?

Tip: When dealing with multiple trigonometric functions, simplifying one step at a time helps avoid calculation mistakes.