Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate to four decimal places, when appropriate. (a) \(\sin (-0.11)\) (b) \(\sec \frac{31}{27}\) (c) \(\tan \left(-\frac{3}{13}\right)\) (d) \(\cos 2.4 \pi\)

Step-by-Step Solution

Answer

(a) -0.1094; (b) 1.1713; (c) -0.2310; (d) -0.3090.

Step 1: Understanding Sine Function

To solve (a), we recognize that the sine function is an odd function, which means that \( \sin(-x) = -\sin(x) \). For small angles, the sine can be approximated directly by using a calculator or appropriate tables. Therefore, \( \sin(-0.11) = -\sin(0.11) \).

Step 2: Calculating Exact Sine Value

Calculate \( \sin(0.11) \) using a calculator, which gives an approximate value of 0.1094. Thus, \( \sin(-0.11) = -0.1094. \)

Step 3: Understanding Secant Function

For (b), recall that secant is the reciprocal of cosine, thus \( \sec(x) = \frac{1}{\cos(x)} \). We need to find \( \sec \left( \frac{31}{27} \right) \).

Step 4: Calculating Cosine and Secant

Calculate \( \cos \left( \frac{31}{27} \right) \) using a calculator. If \( x \) is in radians, this value will be approximately 0.8537. Thus, \( \sec \left( \frac{31}{27} \right) = \frac{1}{\cos \left( \frac{31}{27} \right)} \approx 1.1713. \)

Step 5: Understanding Tangent Function

For (c), use the fact that tangent is an odd function, meaning \( \tan(-x) = -\tan(x) \). Therefore, \( \tan \left(-\frac{3}{13}\right) = -\tan \left(\frac{3}{13}\right). \)

Step 6: Calculating Tangent Value

Calculate \( \tan \left( \frac{3}{13} \right) \) using a calculator. The approximate value is 0.2310. Therefore, \( \tan \left(-\frac{3}{13}\right) = -0.2310. \)

Step 7: Understanding Cosine Function for Circular Angles

For (d), note that \( \cos \) is periodic with a period of \( 2 \pi \). This means \( \cos(2.4\pi) \) is equivalent to \( \cos(2.4\pi - 2\pi) = \cos(0.4\pi) \).

Step 8: Calculating Cosine Value

Calculate \( \cos(0.4\pi) \) using a calculator. This evaluates to approximately -0.3090.

Key Concepts

sine function

The sine function, often denoted as \( \sin(x) \), is one of the primary trigonometric functions. It relates an angle of a right triangle to the ratios of the side lengths. The sine of an angle \( x \) in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

• Key property: Sine is an odd function. This means \( \sin(-x) = -\sin(x) \), which reflects symmetry about the origin.

Using a calculator or trigonometric tables, you can directly find \( \sin(0.11) \). For small angles, the value can be approximated, in this case, as approximately 0.1094. Hence, \( \sin(-0.11) = -0.1094 \). Sine is periodic, repeating every \( 2\pi \) radians, a property that makes it fundamental in modeling wave behaviors.

secant function

The secant function, denoted as \( \sec(x) \), is the reciprocal of the cosine function. This means \( \sec(x) = \frac{1}{\cos(x)} \). It is defined for all angles where the cosine is not zero, and therefore it extends beyond the typical bounds of sine and cosine.

• The secant function can be used to derive several properties and trigonometric identities.

In exploring \( \sec \left( \frac{31}{27} \right) \), the first step is to find \( \cos \left( \frac{31}{27} \right) \) using a calculator, yielding a value of approximately 0.8537. Consequently, \( \sec \left( \frac{31}{27} \right) = \frac{1}{0.8537} \approx 1.1713 \). Secant arises in various applications, particularly in calculus and analytic geometry, enhancing its mathematical significance.

tangent function

The tangent function, written as \( \tan(x) \), plays a crucial role in trigonometry. It is defined as the ratio of the sine and cosine functions: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This makes it central to many trigonometric identities and relationships.

• Tangent is also an odd function, meaning \( \tan(-x) = -\tan(x) \).

To find \( \tan \left(-\frac{3}{13}\right) \), first calculate \( \tan \left(\frac{3}{13}\right) \), which approximates to 0.2310. Then, applying the odd function property, \( \tan \left(-\frac{3}{13}\right) = -0.2310 \).The periodicity of tangent is different from sine and cosine; it repeats every \( \pi \) radians. This property is frequently used in solving trigonometric equations.

cosine function

The cosine function, denoted as \( \cos(x) \), is vital in trigonometry for relating the adjacent side of a right triangle to the hypotenuse. Like sine, cosine has a crucial role in defining angles and cycles.

• Unlike the sine function, cosine is an even function, so \( \cos(-x) = \cos(x) \).
• Cosine has a periodicity of \( 2\pi \) radians.

For \( \cos(2.4\pi) \), we leverage its periodic nature. By reducing \( 2.4\pi \) by \( 2\pi \), we handle the angle as \( 0.4\pi \), simplifying the calculation. Using a calculator, \( \cos(0.4\pi) \) yields approximately -0.3090.Cosine graphs are symmetric about the y-axis, making them particularly useful in physics and engineering to model oscillations and waves.