Question

If \(x=4\) tan \(\theta,\) express \(\sin \theta\) in terms of \(x\)

Short answer

In this problem, we were given that \(x=4\text{tan}\ \theta\) and were asked to express \(\sin\theta\) in terms of \(x\). Through a series of steps using trigonometric identities and algebraic manipulations, we were able to find the expression: \(\sin \theta = \frac{x}{\sqrt{x^2 + 16}}\).

Step-by-step solution

Express \(\tan \theta\) in terms of \(x\)

We are given the equation \(x=4 \text{tan} \ \theta\). To express \(\tan \theta\) in terms of \(x\), divide both sides by 4: \(\text{tan} \ \theta = \frac{x}{4}\)

Recall the definition of \(\tan \theta\)

The tangent function is defined as the ratio of the sine function to the cosine function: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

Substitute the expression for \(\tan \theta\) in terms of \(x\)

Replace \(\tan \theta\) in the definition with the expression we found in Step 1: \(\frac{x}{4} = \frac{\sin \theta}{\cos \theta}\)

Find an expression for \(\sin \theta\)

To find an expression for \(\sin \theta\), we can multiply both sides of the equation by \(\cos \theta\): \(\sin \theta = \frac{x}{4} \cos \theta\)

Use the Pythagorean identity

Recall the Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\) Since we have an expression for \(\sin \theta\) in terms of \(x\) and \(\cos \theta\), we can square both sides of the equation found in Step 4: \((\sin \theta)^2 = (\frac{x}{4}\cos \theta)^2\)

Replace \((\sin \theta)^2\) in the Pythagorean identity and solve for \(\cos^2\theta\)

Replace \((\sin \theta)^2\) in the Pythagorean identity with \((\frac{x}{4}\cos \theta)^2\): \((\frac{x}{4}\cos \theta)^2 + \cos^2\theta = 1\) Now, factor out \(\cos^2\theta\): \(\cos^2\theta (\frac{x^2}{16} + 1) = 1\) To find \(\cos^2\theta\), divide both sides by \((\frac{x^2}{16} + 1)\): \(\cos^2\theta = \frac{1}{\frac{x^2}{16} + 1}\)

Find \(\sin^2\theta\) using the Pythagorean identity

Using the Pythagorean identity and the expression for \(\cos^2\theta\) we found in Step 6: \(\sin^2\theta = 1 - \cos^2\theta\) Substitute the expression for \(\cos^2\theta\): \(\sin^2\theta = 1 - \frac{1}{\frac{x^2}{16} + 1}\)

Solve for \(\sin \theta\)

To find \(\sin \theta\), we take the square root of both sides of the equation. Keep in mind that taking a square root will give two different values, positive and negative. However, we will only consider the positive value here for simplicity: \(\sin \theta = \sqrt{1 - \frac{1}{\frac{x^2}{16} + 1}}\)

Simplify the expression

To simplify the expression for \(\sin \theta\), find a common denominator: \(\sin \theta = \sqrt{\frac{(\frac{x^2}{16} + 1) - 1}{\frac{x^2}{16} + 1}}\) Simplify the numerator: \(\sin \theta = \sqrt{\frac{\frac{x^2}{16}}{\frac{x^2}{16} + 1}}\) Finally, multiply both the numerator and denominator by 16 to get rid of the fractions: \(\sin \theta = \frac{x}{\sqrt{x^2 + 16}}\) Now we have expressed \(\sin \theta\) in terms of \(x\).

Key concepts

Tangent Function

In trigonometry, the tangent function, represented as \( \tan \theta \), plays a crucial role in relating the sine and cosine functions. It is defined as the ratio of the length of the opposite side to the adjacent side in a right-angled triangle. Mathematically, it is expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This relationship is fundamental to understanding triangles and their properties.

• This equation tells us that if you know the values of sine and cosine for a particular angle, you can easily find the tangent.
• This property of tangent is extensively used in solving trigonometrical expressions where direct measurement might not be possible.

In our exercise, we transformed the equation \( x = 4 \tan \theta \) to express \( \tan \theta = \frac{x}{4} \). This simple transformation makes it easier for us to manipulate the expression further and relate it to other trigonometric identities.

Pythagorean Identity

The Pythagorean Identity is a vital aspect of trigonometry that forms the backbone of many trigonometric relationships. It states that for any angle \( \theta \), the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) always holds true. This identity arises from the Pythagorean theorem in a unit circle, where the radius is 1.

• This identity allows us to express one trigonometric function in terms of another, thereby simplifying complex expressions.
• It also helps in proving other trigonometric equations and relationships.

In our given problem, we used this identity to find \( \cos^2 \theta \) when we already know the expression for \( \sin \theta \) in terms of \( x \). By manipulating this identity, we were able to arrive at expressions that describe the trigonometric relationships of known quantities.

Trigonometric Ratios

Trigonometric ratios are simple relationships between the angles and sides of a right-angled triangle. These ratios are fundamental in solving problems related to angles and distances. The primary trigonometric ratios are sine, cosine, and tangent. These are defined as:

• Sine (\(\sin \theta\)): the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\(\cos \theta\)): the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\(\tan \theta\)): the ratio of the length of the opposite side to the adjacent side.

Knowing these ratios helps in deriving other trigonometric identities and expressions. For example, in our problem, we needed to convert \( \tan \theta = \frac{x}{4} \) into related expressions of \( \sin \theta \) using the identities mentioned.
Understanding these basic ratios not only aids in solving elementary trigonometric problems but also provides the building blocks for more advanced mathematical concepts, such as calculus and analytical geometry. By manipulating these ratios and identities, one can solve a vast array of mathematical challenges, reinforcing their significance in the mathematical toolkit.