Trigonometric Ratios

Trigonometry is all about the relationship between angles and the sides of a right triangle. There are six key ratios to remember:

1. Sine (sin) = opposite/hypotenuse
2. Cosine (cos) = adjacent/hypotenuse
3. Tangent (tan) = opposite/adjacent
4. Secant (sec) = hypotenuse/adjacent (reciprocal of cosine)
5. Cosecant (csc) = hypotenuse/opposite (reciprocal of sine)
6. Cotangent (cot) = adjacent/opposite (reciprocal of tangent)

Angles can be measured in degrees or radians. Degrees are familiar - a full circle equals 360°. Radians are based on the radius of a circle, with a full circle equal to 2π radians (about 6.28). To convert between them:

• Degrees to radians: multiply by π/180
• Radians to degrees: multiply by 180/π

💡 A quick conversion shortcut: 180° equals π radians, so 90° equals π/2 radians, 60° equals π/3 radians, etc.

For example, to convert 135° to radians, you calculate: (135° × π)/180 = 3π/4 radians.

Angles on the Coordinate Plane

When graphing angles on a coordinate plane, you always start with the initial ray on the positive x-axis. The angle opens counterclockwise for positive angles and clockwise for negative ones.

The coordinate plane has four quadrants, each with unique angle ranges:

• Quadrant I: 0° to 90° $0 to π/2 radians$
• Quadrant II: 90° to 180° $π/2 to π radians$
• Quadrant III: 180° to 270° $π to 3π/2 radians$
• Quadrant IV: 270° to 360° $3π/2 to 2π radians$

Coterminal angles have the same terminal rays. If two angles differ by a multiple of 360° (or 2π radians), they're coterminal.

Arc Length and Sector Area

When an angle opens in a circle, it creates an arc and a sector. The formulas to remember:

Arc length (s) with radius (r) and angle θ:

• In radians: s = r × θ
• In degrees: s = r × θ × (π/180)

Sector area with radius (r) and angle θ:

• In radians: A = (θ × r²)/2
• In degrees: A = (θ/360) × π × r²

🔍 Using radians makes the formulas simpler! That's why mathematicians and physicists prefer radians over degrees.

Coordinate Trigonometry and Positivity

In the coordinate plane, the trigonometric functions work for any angle, not just in right triangles. Each quadrant has different positive and negative values:

• Quadrant I (0°-90°): All trig functions are positive
• Quadrant II (90°-180°): Only sine and cosecant are positive
• Quadrant III (180°-270°): Only tangent and cotangent are positive
• Quadrant IV (270°-360°): Only cosine and secant are positive

A helpful memory trick: "All Students Take Calculus" - moving clockwise from Quadrant I: All, Sine, Tangent, Cosine.

Special Angles and Triangles

The unit circle $radius = 1$ helps us find exact values for trig functions. Two special right triangles are particularly useful:

1. 30°-60°-90° triangle: If the shortest leg is x, then the other leg is x√3 and the hypotenuse is 2x.

2. 45°-45°-90° triangle: Both legs are equal (x), and the hypotenuse is x√2.

The reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps find trig values for angles in any quadrant.

🧠 Understanding the unit circle is like having a cheat sheet for trigonometry! It connects coordinate geometry with trigonometric values.

Trigonometric Functions and Their Graphs

Trig functions create distinctive wave patterns when graphed. The sine function, f(x) = sin(x), has these key characteristics:

• Domain: All real numbers
• Range: Values between -1 and 1
• Central axis: y = 0 (the middle line of the wave)
• Amplitude: 1 (distance from central axis to peak)
• Period: 2π (length of one complete cycle)
• Frequency: 1/(2π) (cycles per unit)

For the general form f(x) = A·sin$Bx+C$+D:

• |A| is the amplitude
• 2π/B is the period
• D shifts the function vertically

Cosine graphs are similar to sine graphs but shifted horizontally. If a graph starts at its middle value, it's a sine function. If it starts at a maximum or minimum, it's a cosine function.

Tangent, cotangent, secant, and cosecant have different graphs with vertical asymptotes - vertical lines the graphs approach but never touch. For example, tan(x) has asymptotes at x = π/2 + nπ (where n is any integer).

🌊 Think of sine and cosine as smooth ocean waves, while tangent and the other functions are like mathematical tsunamis that shoot toward infinity!

Other Trigonometric Function Graphs

The remaining trig functions have unique graphing characteristics:

Cotangent function f(x) = cot(x):

• Domain: All real numbers except x = nπ (where n is any integer)
• Range: All real numbers
• Has vertical asymptotes at x = nπ
• Period of π

Secant function f(x) = sec(x):

• Domain: All real numbers except x = (π/2) + nπ
• Range: Values less than or equal to -1 or greater than or equal to 1
• Has vertical asymptotes at x = (π/2) + nπ
• Period of 2π

Cosecant function f(x) = csc(x):

• Domain: All real numbers except x = nπ
• Range: Values less than or equal to -1 or greater than or equal to 1
• Has vertical asymptotes at x = nπ
• Period of 2π

📊 Notice how secant is related to cosine, and cosecant is related to sine! Sec(x) = 1/cos(x) and csc(x) = 1/sin(x), which explains their similar periods but different asymptotes.