Question

θ is an acute angle and sin θ and cos θ are given. use identities to find tan θ, csc θ, sec θ, and cot θ. where necessary, rationalize denominators. sin θ = \frac{24}{25}, cos θ = \frac{7}{25} tan θ = \square (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Recall the tangent identity

The identity for tangent is \(\tan\theta=\frac{\sin\theta}{\cos\theta}\).

Step2: Substitute the given values

We know that \(\sin\theta = \frac{24}{25}\) and \(\cos\theta=\frac{7}{25}\). Substituting these into the formula, we get \(\tan\theta=\frac{\frac{24}{25}}{\frac{7}{25}}\).

Step3: Simplify the fraction

When dividing fractions, we multiply by the reciprocal, so \(\frac{\frac{24}{25}}{\frac{7}{25}}=\frac{24}{25}\times\frac{25}{7}\). The 25 in the numerator and denominator cancels out, leaving \(\frac{24}{7}\).

Answer:

\(\frac{24}{7}\)