Problem 44 - Algebra and Trigonometry

Question

Use a half-angle formula to find the exact value of each expression. $$ \tan 112.5^{\circ} $$

Step-by-Step Solution

Answer

\(\tan(112.5^{\circ}) = 1 + \sqrt{2}\)

Step 1: Express Given Angle as Half of Another Known Angle

Express 112.5 degrees as half of 225 degrees because 225 degrees is a known angle on the unit circle. So, we have \(x/2 = 112.5^{\circ}\) and \(x = 225^{\circ}\).

Step 2: Identify the Cosine Value of the New Angle

Identify the cosine of 225 degrees. The coordinates for 225 degrees on the unit circle are \((- \sqrt{2}/2, - \sqrt{2}/2)\), so \(\cos 225^{\circ} = - \sqrt{2}/2.\)

Step 3: Apply the Half-Angle Formula to Calculate

Using the half-angle formula for tangent \(\tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}\), and \(\cos^{2} 225 = 2\), we substitute these values into the formula and get \(\tan(112.5^{\circ}) = \sqrt{\frac{1 - (-\sqrt{2}/2)}{1 + (-\sqrt{2}/2)}} = \sqrt{\frac{1 + \sqrt{2}/2}{1 - \sqrt{2}/2}}\). Simplifying further, we obtain \(\tan(112.5^{\circ}) = 1 + \sqrt{2}\).

Key Concepts

Tangent

Tangent is one of the primary trigonometric functions used in math, specifically in conjunction with angles and the unit circle. It is defined as the ratio of the sine and cosine of an angle.

• The tangent of an angle \(\theta\), is given by \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Understanding tangent is crucial when dealing with angles, as it provides insights into the relationship between the sides of a right triangle. In the context of the unit circle, tangent represents the slope of the line that forms the angle from the origin. The positive or negative value of the tangent depends on the quadrant in which the angle is located.For example, in this exercise, we wanted to calculate \(\tan 112.5^\circ\). Using the half-angle formula allows us to determine this by expressing 112.5 degrees as half of 225 degrees. The tangent at 112.5 degrees comes out to be \(1 + \sqrt{2}\), which provides the exact value using trigonometric identities.

Unit Circle

The unit circle is a fundamental concept in trigonometry used to understand the properties of angles and trigonometric functions. It is a circle with a radius of one, centered at the origin of a coordinate plane.

• Coordinates on this circle are essential as they represent both sine and cosine values for the respective angles.
• Angles are measured in either degrees or radians, and the position of a point on the circle helps us find associated trigonometric function values.

For instance, in our exercise with 112.5 degrees, we considered the unit circle to find a more easily recognizable angle, 225 degrees, whose coordinates \((- \sqrt{2}/2, - \sqrt{2}/2)\) provide known sine and cosine values.The unit circle helps systematically determine trigonometric values when angles are beyond typical right triangles, supporting approximations and simplifying calculations with angles measured in radians and any circular motion.

Cosine

Cosine is another essential trigonometric function. For a given angle \(\theta\), cosine is the x-coordinate of the corresponding point on the unit circle.

• This means for any point \((x, y)\) on the unit circle, \(\cos \theta = x\).
• Cosine values can be positive or negative, depending upon the angle's quadrant, with specific values at key angles such as 0, 90, 180, and 270 degrees.

In solving for \(\tan 112.5^\circ\), determining \(\cos 225^\circ\) was essential. Since 225 degrees lies in the third quadrant of the unit circle, both sine and cosine are negative. Therefore, \(\cos 225^\circ = -\sqrt{2}/2\).Knowing the cosine allowed us to effectively use the half-angle formula to derive the tangent of our desired angle.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold for all values of the included variables. These identities are crucial for simplifying expressions and solving trigonometric problems.

• Common identities include Pythagorean identities, angle sum and difference identities, and half-angle formulas.
• Half-angle formulas help compute the trigonometric functions for angles that are half of known angles.

In our task of solving \(\tan 112.5^\circ\), the half-angle formula was indispensable. The formula for tangent at half an angle is \(\tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}\).By substituting \(\cos 225^\circ\) into the half-angle formula, it allowed us to find the exact value: \(1 + \sqrt{2}\). This use of trigonometric identities not only simplifies complex problems but ensures accuracy in calculations.