Problem 33 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \sin 0^{\circ}+\cos 0^{\circ}+\tan 0^{\circ} $$

Step-by-Step Solution

Answer

The exact value is 1.

Step 1: Identify Trigonometric Values at 0 Degrees

First, we find the trigonometric values at 0 degrees. We know from trigonometric properties:\(\sin 0^{\circ} = 0\), \(\cos 0^{\circ} = 1\), and \(\tan 0^{\circ} = 0\) because \(\tan 0^{\circ} = \frac{\sin 0^{\circ}}{\cos 0^{\circ}} = \frac{0}{1} = 0\).

Step 2: Substitute Values into the Expression

Using the trigonometric values from Step 1, substitute them into the given expression:\[\sin 0^{\circ} + \cos 0^{\circ} + \tan 0^{\circ} = 0 + 1 + 0\]

Step 3: Add the Values Together

Now, add the substituted values together:\[0 + 1 + 0 = 1\]

Key Concepts

Sine Function

The sine function is one of the foundational trigonometric functions and is often abbreviated as "sin." It expresses the ratio of the length of the side opposite to the angle in a right triangle to the length of the hypotenuse. For an angle \(\theta\) in a standard Cartesian coordinate system, the sine of \(\theta\) can also be understood as the y-coordinate of a point on the unit circle, which is a circle with a radius of 1 centered at the origin.

• The function is periodic with a period of \(360^\circ\) or \(2\pi\) radians.
• It starts at 0, rises to 1, then returns to 0 and decreases to -1, and then back to 0, forming a wave-like pattern.

In this exercise, it was noted that \(\sin 0^\circ = 0\). This means that the sine of 0 degrees is 0 because the opposite side does not exist at that angle, making the ratio 0.

Cosine Function

The cosine function, abbreviated as "cos," is a counterpart to the sine function and also a fundamental trigonometric function. It measures the ratio of the length of the adjacent side of an angle in a right triangle to the hypotenuse. Like the sine function, it can be visualized using the unit circle, where it represents the x-coordinate of a point corresponding to an angle \(\theta\).

• Cosine is periodic with the same period as sine: \(360^\circ\) or \(2\pi\) radians.
• It starts at 1, decreases to 0, then -1, back to 0, and up to 1, creating a wave-like pattern.

In the solution, \(\cos 0^\circ = 1\), indicating that at 0 degrees, the x-coordinate of the point on the unit circle is at its maximum value, 1.

Tangent Function

The tangent function, or "tan," is unique among the trigonometric functions as it is not directly represented by the sides of a triangle in the unit circle model. Instead, tangent is defined as the ratio of the sine of an angle to the cosine of the same angle, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

• Tangent has a period of \(180^\circ\) or \(\pi\) radians, unlike sine and cosine.
• The function is undefined whenever the cosine of the angle is zero, resulting in vertical asymptotes.

For this angle, \(\tan 0^\circ = \frac{0}{1} = 0\), showing that the sine divided by cosine at this point is 0, as the sine is 0.

Trigonometric Identities

Trigonometric identities are mathematical equations expressing relationships between trigonometric functions. They are useful for simplifying expressions and solving equations.One of the key identities is the Pythagorean identity:\[\sin^2 \theta + \cos^2 \theta = 1\]This indicates that for any angle \(\theta\), the sum of the squares of sine and cosine equals 1.There are also angle sum and difference identities, double angle identities, and others that describe how these functions interact under various conditions. These identities are essential for proving more complex trigonometric equations and solving problems just like the given exercise, where understanding the relationships between sine, cosine, and tangent helped determine the values at 0 degrees.