Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-10: Find the exact values. (a) \(\tan 60^{\circ}+\tan 225^{\circ}\) (b) \(\tan 285^{\circ}\) (use \(285^{\circ}=60^{\circ}+225^{\circ}\) )

Step-by-Step Solution

Answer

(a) \(\sqrt{3} + 1\), (b) \(-3 - \sqrt{3}\).

Step 1: Identify Known Tangent Values

First, recall the known tangent values of standard angles: - \( \tan 60^{\circ} = \sqrt{3} \) since it is the tangent of an angle in a 30-60-90 triangle. - \( \tan 225^{\circ} = 1 \) because it is equivalent to \( \tan 45^{\circ} \), but in the third quadrant where tangent remains positive.

Step 2: Solve (a) \(\tan 60^{\circ} + \tan 225^{\circ}\)

With the values found in step 1, calculate the sum: \(\tan 60^{\circ} + \tan 225^{\circ} = \sqrt{3} + 1.\)

Step 3: Use Angle Addition Formula for Tangent

To find \(\tan 285^{\circ}\) using \(285^{\circ} = 60^{\circ} + 225^{\circ}\), apply the tangent addition formula:\[\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\]Here \(A = 60^{\circ}\) and \(B = 225^{\circ}\).

Step 4: Substitute Values into Formula

Substitute the values of \(\tan 60^{\circ} = \sqrt{3}\) and \(\tan 225^{\circ} = 1\) into the formula:\[\tan 285^{\circ} = \frac{\sqrt{3} + 1}{1 - (\sqrt{3} \cdot 1)} = \frac{\sqrt{3} + 1}{1 - \sqrt{3}}.\]

Step 5: Simplify the Expression

To simplify the expression \(\frac{\sqrt{3} + 1}{1 - \sqrt{3}}\), multiply the numerator and the denominator by the conjugate of the denominator:\[\frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{-2}.\]

Step 6: Final Calculation

Calculate the final simplified form:The numerator becomes:\[(\sqrt{3} + 1)(1 + \sqrt{3}) = \sqrt{3} + 3 + \sqrt{3} + \sqrt{3}\cdot \sqrt{3} = \sqrt{3} + 3 + \sqrt{3} + 3 = 6 + 2\sqrt{3}.\]Thus, \[\tan 285^{\circ} = \frac{6 + 2\sqrt{3}}{-2} = -3 - \sqrt{3}.\]

Key Concepts

Angle Addition Formula

The angle addition formula is a key tool when working with trigonometric functions, especially for finding the tangent of a sum of angles. When you have two angles, say \(A\) and \(B\), the tangent addition formula helps to find the tangent of the angle formed by adding \(A\) and \(B\). It is written as follows:

• \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)

This formula essentially derives from the sine and cosine angle addition formulas but is rearranged specifically for tangent.
To use it effectively, you need to know the tangent values of the individual angles.
In the exercise example, to find \(\tan 285^{\circ}\), we recognize that this can be expressed as \(60^{\circ} + 225^{\circ}\). Using the angle addition formula simplifies the process to find the tangent of the new angle.

Trigonometric Identities

Trigonometric identities are fundamental properties connecting the trigonometric functions. They provide relationships between these functions that are true for all angle measures.
There are several types of identities, such as Pythagorean identities, reciprocal identities, and co-function identities. Here, we focus on the angle addition identities and specifically, the tangent addition formula mentioned earlier.

• Pythagorean identities, like \( \sin^2\theta + \cos^2\theta = 1 \), aren't directly used in this problem but are essential in understanding why these identities hold true.
• Tangent identities help us combine and manipulate angles, facilitating the calculation of non-standard angles like \(285^{\circ}\).

By substituting known values directly into these identities, we can solve complex trigonometric equations readily.

Standard Angle Values

Understanding standard angle values is crucial while working with trigonometric tasks, as these are the angles where the trigonometric functions have known exact values.
Common standard angles include \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ},\) and \(90^{\circ}\), and their respective radian measures.

• The tangent of \(60^{\circ}\) is \(\sqrt{3}\), derived from the special 30-60-90 triangle properties.
• Similarly, \(\tan 225^{\circ} = 1\) because it relates to \(\tan 45^{\circ}\), and since \(225^{\circ}\) is in the third quadrant where tangent remains positive.

Memorizing these values can significantly ease the process of solving trigonometric problems, as you will instantly recognize angles and apply identities without delay. In exercises like finding \(\tan 285^{\circ}\), these standard values make the calculations straightforward and manageable.