Trigonometric Identities, Simplification, and Triangle Solving

Trigonometric Identities and Simplification

Proving Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which both sides are defined. Proving these identities often involves algebraic manipulation and the use of fundamental trigonometric relationships.

• Example: Prove the following identity: $\csc 2x + \cot 2x = \cot x$

• Key Steps:

  • Recall that $\csc \theta = \frac{1}{\sin \theta}$ and $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

  • Use double angle identities: $\sin 2x = 2\sin x \cos x$ and $\cos 2x = \cos^2 x - \sin^2 x$.

  • Express all terms in terms of $\sin x$ and $\cos x$ and simplify.

Simplifying Trigonometric Expressions

Simplification often involves using angle addition or subtraction formulas, and recognizing standard angle values.

• Example: Simplify $\cos\left(x + \frac{\pi}{2}\right)$

• Key Formula: $\cos(A + B) = \cos A \cos B - \sin A \sin B$

• Application: $\cos\left(x + \frac{\pi}{2}\right) = \cos x \cos \frac{\pi}{2} - \sin x \sin \frac{\pi}{2} = 0 - \sin x = -\sin x$

Exact Values of Trigonometric Functions

Evaluating Sine and Cosine at Special Angles

Some trigonometric values can be found exactly using known values for special angles and the unit circle.

• Example: Find the exact value of $\sin\left(-\frac{7\pi}{12}\right)$

• Key Steps:

  • Express $-\frac{7\pi}{12}$ as a sum or difference of known angles, e.g., $-\frac{7\pi}{12} = -\frac{3\pi}{4} + \frac{\pi}{3}$.

  • Use the sine addition/subtraction formula: $\sin(A - B) = \sin A \cos B - \cos A \sin B$.

  • Substitute known values for $\sin$ and $\cos$ at $\frac{3\pi}{4}$ and $\frac{\pi}{3}$.

Double Angle Formulas

The double angle formulas allow us to find the sine or cosine of twice an angle in terms of the original angle.

• Formulas:

  • $\sin 2x = 2 \sin x \cos x$

  • $\cos 2x = \cos^2 x - \sin^2 x$

• Example: Given $\sin x = \frac{3}{8}$ and $0 < x < \frac{\pi}{2}$, find $\sin 2x$.

• Key Steps:

  • Find $\cos x$ using $\sin^2 x + \cos^2 x = 1$.

  • Calculate $\sin 2x = 2 \sin x \cos x$.

Solving Triangles

Solving a Triangle Given Three Sides (SSS Case)

When all three sides of a triangle are known, the triangle can be solved using the Law of Cosines and the Law of Sines.

• Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos C$

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

• Example: Given $a = 9$, $b = 7$, $c = 5$, find all angles $A$, $B$, $C$.

• Key Steps:

  • Use the Law of Cosines to find one angle, e.g., $A$.

  • Use the Law of Sines to find another angle.

  • Find the third angle using the fact that the sum of angles in a triangle is $180^\circ$ or $\pi$ radians.

Summary Table: Key Trigonometric Formulas

Formula	Description
$\sin^2 x + \cos^2 x = 1$	Pythagorean Identity
$\sin 2x = 2 \sin x \cos x$	Double Angle Formula for Sine
$\cos 2x = \cos^2 x - \sin^2 x$	Double Angle Formula for Cosine
$\cos(A + B) = \cos A \cos B - \sin A \sin B$	Cosine Addition Formula
$\sin(A - B) = \sin A \cos B - \cos A \sin B$	Sine Subtraction Formula
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$	Law of Sines
$c^2 = a^2 + b^2 - 2ab \cos C$	Law of Cosines