Calculating Trigonometric Values and Simplifying Expressions

In this article, we will go through the steps to calculate trigonometric values and simplify expressions involving tangent, sine, and cosine. We will use the given information about the tangent to find the sine and cosine, and then apply these values to simplify a complex expression. This content is designed to be SEO-friendly for Google and includes detailed explanations with relevant keywords and structured content.

Understanding the Problem

Given that tan x 8/15, we start by recognizing that tangent is the ratio of the sine to the cosine (tan x sin x / cos x). This information allows us to construct a right triangle where the opposite side is 8 and the adjacent side is 15. The hypotenuse can be found using the Pythagorean theorem:

[text{Hypotenuse} sqrt{8^2 15^2} sqrt{64 225} sqrt{289} 17]

Calculating Sine and Cosine

Using the right triangle properties, we can identify:

[sin x frac{8}{17}]

[cos x frac{15}{17}]

Since we have two possible quadrants for the angle x (1st or 3rd), we consider both scenarios:

Quadrant 1: Both sine and cosine are positive. [sin x frac{8}{17}] [cos x frac{15}{17}] Quadrant 3: Both sine and cosine are negative. [sin x -frac{8}{17}] [cos x -frac{15}{17}]

Simplifying the Expression

We are asked to simplify the expression [frac{sin x cdot cos x}{cos x (1 - cos x)}]

Let's simplify this step by step, starting by substituting the values of (sin x) and (cos x).

First Quadrant: Substitute (sin x frac{8}{17}) and (cos x frac{15}{17}) The expression becomes: [frac{frac{8}{17} cdot frac{15}{17}}{frac{15}{17} (1 - frac{15}{17})}] Simplify the numerator and the denominator: [frac{8 cdot 15 / 289}{frac{15}{17} cdot frac{2}{17}}] Further simplification yields: [frac{120}{289} cdot frac{289}{30} frac{120}{30} 4] Third Quadrant: Substitute (sin x -frac{8}{17}) and (cos x -frac{15}{17}) The expression becomes: [frac{-frac{8}{17} cdot -frac{15}{17}}{-frac{15}{17} (1 frac{15}{17})}] Simplify the numerator and the denominator: [frac{8 cdot 15 / 289}{-frac{15}{17} cdot frac{32}{17}}] Further simplification yields: [frac{120}{289} cdot frac{289}{30} frac{120}{30} 4]

Therefore, the final answer is:

[boxed{frac{391}{30}, frac{391}{480}}]

Additional Tips

To solve similar trigonometric problems, it's helpful to:

Draw a right triangle to visualize the given values. Use the SOHCAHTOA mnemonic to remember the definitions of sine, cosine, and tangent. Apply the Pthagorean theorem to find unknown sides in the right triangle. Simplify expressions step by step, ensuring you keep track of the signs and fractions.

For more trigonometry practice and guidance, consider using online resources or textbooks that cover the basics of trigonometric functions and identities.