- Understand and use the basic rules and relationships in trigonometry
Trigonometric Identities
Trigonometric identities are the equalities that involve trigonometric functions and are true for any substitution of the variable where the sides of the equation are defined. Mastery of these identities is crucial for solving trigonometric equations, proving other mathematical statements, and is frequently necessary in calculus.
The main trigonometric identities are listed below.
Trigonometric Identities
Reciprocal identities
Pythagorean identities
Addition and subtraction formulas
Double-angle formulas
- Association: Link each identity with a visual cue or a part of the unit circle. For example, the sine function starts at zero and goes up, just like a sine wave starts at the middle and rises.
- Mnemonics: Use phrases to remember relationships, like the mnemonic SOHCAHTOA.
- Repetition: Regularly practice rewriting the identities from memory. Repetition is key to retention.
- Flashcards: Create a set of flashcards with each identity on one side and its name or a key hint on the other.
- Group Study: Discuss and solve problems with peers; explaining concepts to others can reinforce your memory.
When facing a trigonometric identity, verification is key to ensuring the identity holds true for all permissible values of the variable. The process is a methodical one, where you manipulate one side of the equation until it matches the other.
- Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build.
- Look for opportunities to factor expressions, square a binomial, or add fractions.
- Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.
- If these steps do not yield the desired result, try converting all terms to sines and cosines.
Prove the trigonometric identity [latex]1+\tan^2 \theta =\sec^2 \theta[/latex].
Algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.
- Spot Patterns: Look for familiar algebraic cues in the equation, like notable identities or factors.
- Substitute Variables: Temporarily replace trig expressions with a single variable to simplify the equation.
- Solve Algebraically: Treat the simplified equation as you would a standard algebraic one.
- Back-Substitute: Once you’ve solved for the temporary variable, revert to the original trigonometric terms.
- Find Angles: Use inverse functions to solve for the angle, considering the function’s period and domain.
- Verify Solutions: Always check that your solutions satisfy the initial equation.
Higher-Order Partial Derivatives
Consider the function
[latex]f(x,y)=2x^{3}-4xy^{2}+5y^{3}-6xy+5x-4y+12[/latex].
Its partial derivatives are
[latex]\dfrac{\partial f}{\partial x}=6x^{2}-4y^{2}-6y+5[/latex] and [latex]\dfrac{\partial f}{\partial y}=-8xy+15y^{2}-6x-4[/latex].
Each of these partial derivatives is itself a function of two variables, so we can calculate partial derivatives of these functions. Just as with single-variable derivatives, we call these second-order derivatives, third-order derivatives, and so on. In general, they are higher-order partial derivatives.
There are four second-order partial derivatives for any function (provided they all exist):
[latex]\large{\frac{\partial^{2}f}{\partial x^{2}}=\frac{\partial}{\partial x}\left[\frac{\partial f}{\partial x}\right],\ \frac{\partial^{2}f}{\partial x\partial y}=\frac{\partial}{\partial x}\left[\frac{\partial f}{\partial y}\right],\ \frac{\partial^{2}f}{\partial y\partial x}=\frac{\partial}{\partial y}\left[\frac{\partial f}{\partial x}\right],\ \frac{\partial^{2}f}{\partial y^{2}}=\frac{\partial}{\partial y}\left[\frac{\partial f}{\partial y}\right]}.[/latex]
An alternative notation for these is [latex]f_{xx}[/latex], [latex]f_{yx}[/latex], [latex]f_{xy}[/latex], and [latex]f_{yy}[/latex], respectively. Higher-order partial derivatives calculated with respect to different variables, such as [latex]f_{xy}[/latex] and [latex]f_{yx}[/latex], are called mixed partial derivatives.
In [latex]\frac{\partial^{2}f}{\partial y\partial x}[/latex], read the variables from right to left: differentiate first with respect to [latex]x[/latex], then with respect to [latex]y[/latex].
In subscript notation [latex]f_{xy}[/latex], read left to right: differentiate first with respect to [latex]x[/latex], then with respect to [latex]y[/latex].
Notice something interesting in the previous example: [latex]\dfrac{\partial^{2}f}{\partial x\partial y}=\dfrac{\partial^{2}f}{\partial y\partial x}[/latex]. Both mixed partial derivatives equal [latex]-3e^{-3y}+10\sin(2x-5y)[/latex]. Is this always true? Under certain conditions, yes. In fact, this is guaranteed by the following theorem.
equality of mixed partial derivatives (Clairaut's Theorem)
Suppose [latex]f\,(x,\ y)[/latex] is defined on an open disk [latex]D[/latex] that contains the point [latex](a,\ b)[/latex]. If the functions [latex]f_{xy}[/latex] and [latex]f_{yx}[/latex] are continuous on [latex]D[/latex], then [latex]f_{xy}=f_{yx}[/latex].
Clairaut's theorem guarantees that as long as mixed second-order derivatives are continuous, the order in which we differentiate doesn't matter. We can differentiate with respect to [latex]x[/latex] first then [latex]y[/latex], or [latex]y[/latex] first then [latex]x[/latex]—we'll get the same result. This theorem can be extended to higher-order derivatives as well.
Beyond the mixed partial derivatives, we also have [latex]f_{xx}[/latex] (the partial derivative of [latex]f_x[/latex] with respect to [latex]x[/latex]) and [latex]f_{yy}[/latex] (the partial derivative of [latex]f_y[/latex] with respect to [latex]y[/latex]). These aren't mixed—they differentiate with respect to the same variable twice.