Subjects

Trigonometry

Dec 12, 2025

5 pages

Understanding Trigonometric Functions: Basics and Graphs

Gabriella Mulé @bellamule08

Trigonometry connects angles and ratios in right triangles, giving us powerful tools to solve real-world problems. Understanding these... Show more

$trigonometry relationship of the ratio of sides of a right triangle hypotenuse opposite leg adjacent leg -theta sine ratio Sin (0) = $\fr$

Trigonometric Ratios

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

Sine (sin) = opposite/hypotenuse
Cosine (cos) = adjacent/hypotenuse
Tangent (tan) = opposite/adjacent
Secant (sec) = hypotenuse/adjacent (reciprocal of cosine)
Cosecant (csc) = hypotenuse/opposite (reciprocal of sine)
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.

$trigonometry relationship of the ratio of sides of a right triangle hypotenuse opposite leg adjacent leg -theta sine ratio Sin (0) = $\fr$

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.

$trigonometry relationship of the ratio of sides of a right triangle hypotenuse opposite leg adjacent leg -theta sine ratio Sin (0) = $\fr$

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

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

$trigonometry relationship of the ratio of sides of a right triangle hypotenuse opposite leg adjacent leg -theta sine ratio Sin (0) = $\fr$

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!

$trigonometry relationship of the ratio of sides of a right triangle hypotenuse opposite leg adjacent leg -theta sine ratio Sin (0) = $\fr$

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.

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Subjects

Trigonometry

•

Dec 12, 2025

•

5 pages

Understanding Trigonometric Functions: Basics and Graphs

Gabriella Mulé

@bellamule08

Trigonometry connects angles and ratios in right triangles, giving us powerful tools to solve real-world problems. Understanding these relationships helps you calculate distances, heights, and angles that would be impossible to measure directly.

$trigonometry relationship of the ratio of sides of a right triangle hypotenuse opposite leg adjacent leg -theta sine ratio Sin (0) = $\fr$

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Trigonometric Ratios

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

Sine (sin) = opposite/hypotenuse
Cosine (cos) = adjacent/hypotenuse
Tangent (tan) = opposite/adjacent
Secant (sec) = hypotenuse/adjacent (reciprocal of cosine)
Cosecant (csc) = hypotenuse/opposite (reciprocal of sine)
Cotangent (cot) = adjacent/opposite (reciprocal of tangent)

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.

$trigonometry relationship of the ratio of sides of a right triangle hypotenuse opposite leg adjacent leg -theta sine ratio Sin (0) = $\fr$

Sign up to see the contentIt's free!

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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.

$trigonometry relationship of the ratio of sides of a right triangle hypotenuse opposite leg adjacent leg -theta sine ratio Sin (0) = $\fr$

Sign up to see the contentIt's free!

Access to all documents

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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:

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

$trigonometry relationship of the ratio of sides of a right triangle hypotenuse opposite leg adjacent leg -theta sine ratio Sin (0) = $\fr$

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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.

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

$trigonometry relationship of the ratio of sides of a right triangle hypotenuse opposite leg adjacent leg -theta sine ratio Sin (0) = $\fr$

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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.