The Main Idea

Beyond sine and cosine, we often use four other trigonometric functions: tangent, cotangent, secant, and cosecant. These functions are all defined in terms of sine and cosine, which means once you know [latex]\sin(\theta)[/latex] and [latex]\cos(\theta)[/latex], you can find the others. For special angles (30°, 45°, 60°, etc.), these values can be expressed exactly using square roots and fractions, not decimals.

Quick Tips: Exact Values for the Other Trig Functions

1. Definitions (in terms of sine and cosine):

   • • [latex]\tan(\theta) = \dfrac{\sin(\theta)}{\cos(\theta)}[/latex]

     • [latex]\cot(\theta) = \dfrac{\cos(\theta)}{\sin(\theta)}[/latex]

     • [latex]\sec(\theta) = \dfrac{1}{\cos(\theta)}[/latex]

     • [latex]\csc(\theta) = \dfrac{1}{\sin(\theta)}[/latex]

2. Start with Special Angles: Use exact sine and cosine values for [latex]30^\circ[/latex], [latex]45^\circ[/latex], [latex]60^\circ[/latex] (or [latex]\dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{\pi}{3}[/latex]).

   [latex]\begin{aligned}\tan(45^\circ) &= \dfrac{\sin(45^\circ)}{\cos(45^\circ)}\\ &= \dfrac{\dfrac{\sqrt{2}}{2}}{\dfrac{\sqrt{2}}{2}}\\ &= 1 \end{aligned}[/latex]
   [latex]\begin{aligned}\sec(60^\circ) &= \dfrac{1}{\cos(60^\circ)}\\ &= \dfrac{1}{\dfrac{1}{2}}\\ &= 2 \end{aligned}[/latex]
   [latex]\begin{aligned}\csc(30^\circ) &= \dfrac{1}{\sin(30^\circ)}\\ &= \dfrac{1}{\dfrac{1}{2}}\\ &= 2 \end{aligned}[/latex]

3. Undefined Values: Watch for division by zero — if sine or cosine = 0, then csc, sec, tan, or cot may be undefined.

4. Keep It Exact: Always leave answers as simplified fractions or radicals (e.g., [latex]\dfrac{\sqrt{3}}{3}[/latex] instead of decimals).

The Main Idea

Trigonometric functions have symmetry that makes them easier to work with. Some are even functions, which means their graphs are symmetric about the y-axis: [latex]f(-x) = f(x)[/latex]. Others are odd functions, which means their graphs are symmetric about the origin: [latex]f(-x) = -f(x)[/latex].

• Cosine and secant are even functions.

• Sine, tangent, cotangent, and cosecant are odd functions.

This property helps simplify expressions and quickly evaluate trig values for negative angles without needing to sketch the unit circle every time.

Quick Tips: Even and Odd Trig Functions

1. Cosine & Secant (Even):

   • • [latex]\cos(-\theta) = \cos(\theta)[/latex]

     • [latex]\sec(-\theta) = \sec(\theta)[/latex]

2. Sine, Tangent, Cotangent, Cosecant (Odd):

   • • [latex]\sin(-\theta) = -\sin(\theta)[/latex]

     • [latex]\tan(-\theta) = -\tan(\theta)[/latex]

     • [latex]\cot(-\theta) = -\cot(\theta)[/latex]

     • [latex]\csc(-\theta) = -\csc(\theta)[/latex]

3. Shortcut: Negative angles don’t need the unit circle — just apply the even/odd rule.

4. Graph Connection:

   • • Even functions → mirror symmetry across the y-axis.

     • Odd functions → rotate 180° about the origin and the graph looks the same.

5. Memory Trick: “Cosine is the even one; most others are odd.”

The Main Idea

Trigonometric identities are equations that are always true for all values of the variable where both sides are defined. They let us rewrite trig expressions in simpler or more useful forms, making it easier to solve equations, prove relationships, and evaluate functions. The most important group is the fundamental identities, which serve as the “building blocks” for all other identities.

Quick Tips: Fundamental Trig Identities

1. Reciprocal Identities

   [latex]\sin(\theta) = \dfrac{1}{\csc(\theta)}[/latex]	[latex]\csc(\theta) = \dfrac{1}{\sin(\theta)}[/latex]
   [latex]\cos(\theta) = \dfrac{1}{\sec(\theta)}[/latex]	[latex]\sec(\theta) = \dfrac{1}{\cos(\theta)}[/latex]
   [latex]\tan(\theta) = \dfrac{1}{\cot(\theta)}[/latex]	[latex]\cot(\theta) = \dfrac{1}{\tan(\theta)}[/latex]

2. Quotient Identities

   • [latex]\tan(\theta) = \dfrac{\sin(\theta)}{\cos(\theta)}[/latex]

   • [latex]\cot(\theta) = \dfrac{\cos(\theta)}{\sin(\theta)}[/latex]

3. Pythagorean Identities

   • [latex]\sin^2(\theta) + \cos^2(\theta) = 1[/latex]

   • [latex]1 + \tan^2(\theta) = \sec^2(\theta)[/latex]

   • [latex]1 + \cot^2(\theta) = \csc^2(\theta)[/latex]

4. Why They Matter

   • Simplify expressions (turn complicated fractions into simple forms).

   • Prove other identities by substitution.

   • Solve trig equations efficiently.

5. Memory Trick

   • Think “Sine² + Cosine² = 1” as the Pythagorean baseline.

   • Divide by [latex]\cos^2[/latex] to get the tangent identity; divide by [latex]\sin^2[/latex] to get the cotangent identity.

The Main Idea

Not all angles have “nice” exact values like [latex]30^\circ[/latex], [latex]45^\circ[/latex], or [latex]60^\circ[/latex]. For most angles, we rely on a calculator to approximate trig function values. The key is to make sure the calculator is in the correct mode (degrees or radians) and to interpret the decimal output appropriately. Calculators give numerical approximations, but understanding what the output should look like helps catch errors and makes results more meaningful.

Quick Tips: Using a Calculator for Trig Functions

1. Check the Mode First

   • If your angle is in degrees, set the calculator to degree mode.

   • If your angle is in radians, set the calculator to radian mode.

2. Enter the Function Directly

   • Example: [latex]\sin(40^\circ) \approx 0.6428[/latex]

   • Example: [latex]\cos!\left(\dfrac{\pi}{5}\right) \approx 0.8090[/latex]

3. Expect a Decimal

   • Calculator results are approximations unless the angle is a special value.

4. Round Wisely

   • Use 3–4 decimal places unless more precision is needed.

5. Estimate First

   • Compare to a nearby special angle so you know if your answer makes sense.

   • Example: [latex]\sin(40^\circ)[/latex] should be close to [latex]\sin(45^\circ) = \dfrac{\sqrt{2}}{2} \approx 0.7071[/latex].

6. Common Pitfall

   • Wrong mode = wrong answer. If a value looks way off, double-check whether your calculator is in degrees or radians.