Evaluate each expression without using a calculator.
tan (arcsin (3/5) + arccos (5/7))

Verified step by step guidance

1. Identify the angles involved: let \( \alpha = \arcsin\left(\frac{3}{5}\right) \) and \( \beta = \arccos\left(\frac{5}{7}\right) \). We want to find \( \tan(\alpha + \beta) \).
2. Recall the tangent addition formula: \[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]. So, we need to find \( \tan \alpha \) and \( \tan \beta \).
3. From \( \alpha = \arcsin\left(\frac{3}{5}\right) \), use the definition of sine to find the opposite side as 3 and hypotenuse as 5. Use the Pythagorean theorem to find the adjacent side: \[ \text{adjacent} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} \]. Then, calculate \( \tan \alpha = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{\sqrt{16}} \).
4. From \( \beta = \arccos\left(\frac{5}{7}\right) \), use the definition of cosine to find the adjacent side as 5 and hypotenuse as 7. Use the Pythagorean theorem to find the opposite side: \[ \text{opposite} = \sqrt{7^2 - 5^2} = \sqrt{49 - 25} \]. Then, calculate \( \tan \beta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{24}}{5} \).
5. Substitute \( \tan \alpha \) and \( \tan \beta \) into the tangent addition formula and simplify the expression to find \( \tan(\alpha + \beta) \) without using a calculator.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions like arcsin and arccos return an angle whose sine or cosine is a given value. Understanding their ranges and how to interpret these angles is essential for evaluating expressions involving compositions of trig functions.

Trigonometric Addition Formulas

The tangent addition formula, tan(A + B) = (tan A + tan B) / (1 - tan A tan B), allows the evaluation of the tangent of a sum of angles. Applying this formula requires finding the tangent of each individual angle first.

Right Triangle Relationships

Using the given sine or cosine values, one can construct right triangles to find missing sides and other trigonometric ratios like tangent. This geometric approach helps compute exact values without a calculator.