An equation that is true for all values of the variable for which the trigonometric functions are defined.
Trigonometric identities are not equations to be solved. They are tools to simplify expressions or prove equivalences.
Fundamental Trigonometric Identities
Pythagorean Identities
The Pythagorean identities are derived from the Pythagorean theorem applied to the unit circle.
Consider a point \$(x, y)\$ on the unit circle, where the radius is 1. By the Pythagorean theorem:
\$\$x^2 + y^2 = 1^2 = 1\$\$
Since \$x = \cos \theta\$ and \$y = \sin \theta\$, we have:
\$\$ \cos^2 \theta + \sin^2 \theta = 1 \$\$
\$\sin^2 \theta + \cos^2 \theta = 1\$
This identity expresses the relationship between the sine and cosine of an angle.
\$1 + \tan^2 \theta = \sec^2 \theta\$
Dividing the Pythagorean identity by \$\cos^2 \theta\$ gives:
\$\$ \frac{\sin^2 \theta}{\cos^2 \theta} + 1 = \frac{1}{\cos^2 \theta} \$\$
Since \$\frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta\$ and \$\frac{1}{\cos^2 \theta} = \sec^2 \theta\$, we have:
\$\$ 1 + \tan^2 \theta = \sec^2 \theta \$\$
\$1 + \cot^2 \theta = \csc^2 \theta\$
Dividing the Pythagorean identity by \$\sin^2 \theta\$ gives:
\$\$ 1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta} \$\$
Since \$\frac{\cos^2 \theta}{\sin^2 \theta} = \cot^2 \theta\$ and \$\frac{1}{\sin^2 \theta} = \csc^2 \theta\$, we have:
\$\$ 1 + \cot^2 \theta = \csc^2 \theta \$\$
Reciprocal Identities
The reciprocal identities express the relationship between trigonometric functions and their reciprocals.
\$\csc \theta = \frac{1}{\sin \theta}\$
The cosecant is the reciprocal of the sine.
\$\sec \theta = \frac{1}{\cos \theta}\$
The secant is the reciprocal of the cosine.
\$\cot \theta = \frac{1}{\tan \theta}\$
The cotangent is the reciprocal of the tangent.
Quotient Identities
The quotient identities express trigonometric functions as ratios of other functions.
\$\tan \theta = \frac{\sin \theta}{\cos \theta}\$
The tangent is the ratio of the sine to the cosine.
\$\cot \theta = \frac{\cos \theta}{\sin \theta}\$
The cotangent is the ratio of the cosine to the sine.
Simplifying Trigonometric Expressions
Rewriting in Terms of Sine and Cosine
One common strategy for simplifying trigonometric expressions is to rewrite all functions in terms of sine and cosine.
Simplify \$\frac{\sec \theta}{\csc \theta}\$.
- Rewrite in terms of sine and cosine:
- \$\$ \frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta}} \$\$
- Simplify the fraction:
- \$\$ \frac{1}{\cos \theta} \cdot \frac{\sin \theta}{1} = \frac{\sin \theta}{\cos \theta} \$\$
- Recognize the quotient identity:
- \$\$ \frac{\sin \theta}{\cos \theta} = \tan \theta \$\$
So, \$\frac{\sec \theta}{\csc \theta} = \tan \theta\$.
Combining Fractions
When working with trigonometric expressions that involve fractions, it is often necessary to combine them by finding a common denominator.
Simplify \$\sin \theta \cdot \tan \theta + \cos \theta\$.
- Rewrite in terms of sine and cosine:
- \$\$ \sin \theta \cdot \frac{\sin \theta}{\cos \theta} + \cos \theta = \frac{\sin^2 \theta}{\cos \theta} + \frac{\cos \theta}{1} \$\$
- Find a common denominator:
- \$\$ \frac{\sin^2 \theta}{\cos \theta} + \frac{\cos^2 \theta}{\cos \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\cos \theta} \$\$
- Use the Pythagorean identity:
- \$\$ \frac{1}{\cos \theta} = \sec \theta \$\$
So, \$\sin \theta \cdot \tan \theta + \cos \theta = \sec \theta\$.
Proving Trigonometric Identities
Strategies for Proving Identities
- Start with the more complicated side of the identity and simplify it to match the other side.
- Use fundamental identities to rewrite expressions in simpler forms.
- Combine fractions or split fractions as needed.
- Factor expressions or expand products to simplify.
Prove that \$\sin \theta \cdot \tan \theta + \cos \theta = \sec \theta\$.
- Start with the left side:
- \$\$ \sin \theta \cdot \tan \theta + \cos \theta \$\$
- Rewrite in terms of sine and cosine:
- \$\$ \sin \theta \cdot \frac{\sin \theta}{\cos \theta} + \cos \theta = \frac{\sin^2 \theta}{\cos \theta} + \frac{\cos \theta}{1} \$\$
- Find a common denominator:
- \$\$ \frac{\sin^2 \theta}{\cos \theta} + \frac{\cos^2 \theta}{\cos \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\cos \theta} \$\$
- Use the Pythagorean identity:
- \$\$ \frac{1}{\cos \theta} = \sec \theta \$\$
So, \$\sin \theta \cdot \tan \theta + \cos \theta = \sec \theta\$.
- Simplify \$\frac{\sec \theta}{\csc \theta}\$.
- Prove that \$\tan^2 \theta + 1 = \sec^2 \theta\$.
- Simplify \$\sin \theta \cdot \cot \theta + \cos \theta\$.
How do we know that trigonometric identities are true for all values of the variable? What role does proof play in establishing mathematical truth?
