Problem 35 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \sin 90^{\circ}+\cos 0^{\circ}+\tan 45^{\circ} $$

Step-by-Step Solution

Answer

The exact value is 3.

Step 1: Understanding Trigonometric Functions

Firstly, recognize and recall the basic trigonometric values for the angles given. Specifically, we use the fact that \( \sin 90^\circ = 1 \), \( \cos 0^\circ = 1 \), and \( \tan 45^\circ = 1 \).

Step 2: Evaluate Each Trigonometric Function

Calculate each of the given trigonometric expressions: \( \sin 90^\circ = 1 \), \( \cos 0^\circ = 1 \), and \( \tan 45^\circ = 1 \).

Step 3: Sum the Values

Add the values from the previous step: \( 1 + 1 + 1 = 3 \).

Key Concepts

Basic Trigonometric Values

Understanding basic trigonometric values is crucial for solving many problems in trigonometry. There are certain standard angles which provide easy-to-remember trigonometric values:

• The sine of 90° is 1. This means when a right angle is made, the sine function identifies the vertical or 'opposite' length as being equal to the hypotenuse.
• The cosine of 0° is 1. At 0°, the cosine measures how far along the horizontal axis you are, which is all the way to the beginning of the axis.
• The tangent of 45° is 1. This indicates that both the vertical and horizontal lengths are equal, making the slope or tangent 1.

These basic values form the foundation for more complex calculations and are worth remembering.

Angle Measures

Angles help determine the values of trigonometric functions. Most trigonometric problems involve either degrees or radians.

• Degrees are more intuitive and based on dividing a circle into 360 parts. An angle of 90° is a quarter of the circle, 45° is an eighth, and so on.
• Radians provide a more mathematically natural measure, rooted in the properties of circles. For example, 90° is equivalent to \( \frac{\pi}{2} \) radians, and 45° is \( \frac{\pi}{4} \).

Understanding the unit of measurement for your angles ensures you interpret trigonometric functions correctly. Working with degrees is often simpler for basic problems, while radians are useful for calculus and advanced studies.

Trigonometric Identities

Trigonometric identities are equations that are true for all angles, forming the backbone of trigonometric proofs and simplifications.

• Pythagorean Identity: This includes \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity helps in connecting functions for any angle measure.
• Angle Sum and Difference Identities: Functions like sine and cosine can be expressed as the sum or difference, for example, \( \sin(A \pm B) \).
• Reciprocal and Ratio Identities: Involve \( \csc \theta, \sec \theta, \cot \theta \), which are inverses such as \( \csc \theta = \frac{1}{\sin \theta} \).

Mastering these identities acts like a toolbox, enabling you to simplify trigonometric expressions, solve equations, and prove other identities effortlessly.