Problem 110 - Algebra and Trigonometry Real Mathematics, Real People

Question

Explain how to use the figure to derive the Pythagorean identities $$\sin ^{2} \theta+\cos ^{2} \theta=1$$ $$1+\tan ^{2} \theta=\sec ^{2} \theta$$ $$\text { and } 1+\cot ^{2} \theta=\csc ^{2} \theta$$ Discuss how to remember these identities and other fundamental trigonometric identities.

Step-by-Step Solution

Answer

The three Pythagorean identities are derived from the Pythagorean theorem in a unit circle. They are \( \sin^2 \theta + \cos^2 \theta = 1 \), \( 1+ \tan^2 \theta = \sec^2 \theta \) and \( 1+ \cot^2 \theta = \csc^2 \theta \). They can be easily remembered by always keeping in mind the unit circle and the basic definitions of sine, cosine, tangent, secant, cosecant, and cotangent.

Step 1: Understand the Trigonometric Circle

Draw a circle with radius 1, also known as the unit circle. Let \( \theta \) represent the angle formed by the positive x-axis and a line segment from the origin (0, 0) to the point on the circle. The x-coordinate of this point is \( \cos \theta \) and the y-coordinate is \( \sin \theta \).

Step 2: Derive the first Pythagorean Identity

From the definition of the unit circle, using the Pythagorean theorem, we have \( \sin^2 \theta + \cos^2 \theta = 1^2 \) that simplifies to \( \sin^2 \theta + \cos^2 \theta = 1 \). So, we derive the first Pythagorean identity.

Step 3: Derive the second and third Pythagorean Identities

We can divide the first identity by \( \cos^2 \theta \) (make sure \( \cos \theta \neq 0 \)). This gives \( \tan^2 \theta + 1 = \sec^2 \theta \). When we divide the first identity by \( \sin^2 \theta \) (making sure \( \sin \theta \neq 0 \)), we get \( 1 + \cot^2 \theta = \csc^2 \theta \).

Step 4: Remembering the Identities

A good way to remember these identities is to always keep in mind the unit circle and the basic definitions of sine, cosine, tangent, secant, cosecant, and cotangent. Furthermore, remembering that these identities all came from the basic Pythagorean theorem in the unit circle makes it easier to derive if forgotten.

Key Concepts

Trigonometric Circle

The trigonometric circle, commonly known as the unit circle in mathematical contexts, is a fundamental construction in trigonometry. It's a circle with a radius of 1 unit, centered at the origin of a coordinate system. The unit circle is instrumental in defining the trigonometric functions for all angles, not just those in the first quadrant. When an angle is drawn in 'standard position', which means its vertex is at the origin and one side lies along the positive x-axis, its other side intersects the unit circle at a certain point. The coordinates of this point are deeply connected to the trigonometric functions of that angle.

For an angle \( \theta \), the horizontal distance from the origin to the point where the angle's ray intersects the unit circle is the cosine of \( \theta \), expressed as \( \cos \theta \), and the vertical distance is the sine of \( \theta \), \( \sin \theta \). The idea is that as \( \theta \) varies, \( \sin \theta \) and \( \cos \theta \) trace out the unit circle, providing a visual and conceptual way to understand these functions and their relationship to angles and, indeed, the Pythagorean identities.

Unit Circle

The unit circle is not only a model for studying angles and their corresponding trigonometric function values, but it is also the bedrock of the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity arises from the definition of the unit circle, which embodies the Pythagorean Theorem. As all points on the unit circle satisfy the equation \( x^2 + y^2 = 1 \), where \( x \) and \( y \) are the coordinates of a point on the circle, we see that when an angle's terminal side intersects the circle, the \( x \) and \( y \) coordinates are indeed the cosine and sine of the angle, respectively.

This relationship is crucial when visualizing trigonometric concepts and solving problems involving angles. Remembering the unit circle's role in trigonometry helps ground the abstract concepts of these mathematical functions into something more concrete and visual.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the context of the unit circle, trigonometric functions acquire their meaning in terms of the coordinates of points on the circle. Apart from sine and cosine, there are other functions like tangent, cotangent, secant, and cosecant. These can be derived from sine and cosine.

Tangent and Cotangent

Tangent, \( \tan \theta \), is the ratio of the sine to the cosine of an angle and can be thought of as the slope of the line connecting the origin to the point on the unit circle. Cotangent, \( \cot \theta \), is the reciprocal of tangent.

Secant and Cosecant

Secant, \( \sec \theta \), is the reciprocal of the cosine, and cosecant, \( \csc \theta \), is the reciprocal of the sine. These functions extend the relationships found in the Pythagorean identities and can also help in solving a variety of trigonometric equations and problems.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is the foundation for the most basic Pythagorean identity in trigonometry: \( \sin^2 \theta + \cos^2 \theta = 1 \), where \( \theta \) is one of the non-right angles in a right triangle.

Understanding this connection allows us to extend the relationship to other trigonometric functions as seen in the identities \( 1 + \tan^2 \theta = \sec^2 \theta \) and \( 1 + \cot^2 \theta = \csc^2 \theta \). These are derived from dividing the basic identity by either \( \cos^2 \theta \) or \( \sin^2 \theta \) respectively. Recognizing the Pythagorean Theorem as the root of these identities can greatly simplify the process of learning and remembering trigonometry.