Problem 15 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(\tan \theta=\frac{7}{6}\)

Step-by-Step Solution

Answer

The approximate value of \(\theta\) would need to be calculated using a calculator.

Step 1: Identify the function

In this problem, the function identified is tangent, which is represented by \(\tan\). It is given that \(\tan(\theta)=\frac{7}{6}\).

Step 2: Apply the inverse tangent function

To solve for \(\theta\), apply the inverse tangent function to both sides of the equation. This gives \(\theta=\arctan(\frac{7}{6})\).

Step 3: Calculate theta in degrees

Calculate theta by finding the arc tangent of \(\frac{7}{6}\) using a calculator. This should give an approximate value of \(\theta\).

Key Concepts

The Inverse Tangent Function

Understanding the inverse tangent function is crucial when solving equations involving tangent, like \(\tan \theta = \frac{7}{6}\).

The tangent function, which in basics relates the ratio of the opposite side to the adjacent side of a right-angled triangle, has an inverse, known as the arc tangent or inverse tangent, represented as \(\arctan\). When you see \(\tan (\theta) = y\), the inverse tangent \(\arctan(y)\) lets you find the angle \theta that would give you the tangent value y.

Applying the inverse tangent to both sides of the original equation allows you to solve for \theta directly. Hence \(\arctan(\tan(\theta)) = \arctan\left(\frac{7}{6}\right)\), which simplifies to \(\theta = \arctan\left(\frac{7}{6}\right)\). Always ensure your calculator is set to the correct mode (degrees or radians) when performing these calculations.

Trigonometric Identities

One cannot overlook the importance of trigonometric identities while dealing with trigonometric equations.

These identities are equations that hold true for all values and are crucial tools in simplifying and solving trigonometry problems. Some common trigonometric identities include the Pythagorean identities, angle sum and difference identities, and reciprocal identities.

For example, the Pythagorean identity \(\tan^2(\theta) + 1 = \sec^2(\theta)\) relates the tangent of an angle to its secant. Understanding and knowing how to apply these identities can help simplify complex tangential equations and can assist in verifying solutions you may reach during your calculations.

Theta Calculation

The process of theta calculation typically involves finding the value of the angle \theta that satisfies a given trigonometric equation.

In our example where \(\tan(\theta) = \frac{7}{6}\), after applying the inverse tangent function, you need to find the numerical value of \theta. This is usually done with a scientific calculator. Given that the range of the inverse tangent function is between \( -90\degree \), and \( 90\degree \), the output for \theta will fall within this range.

It is necessary to be aware of the quadrants in which your angle \theta may lie because the tangent function is positive in the first and third quadrants. Knowing the context can help predict if we need to add \(180\degree\) to our result to get the correct angle as per the given situation.