What Is Trigonometry? Formulas, Examples, Applications and Tables

What is trigonometry? It is the branch of mathematics that studies the relationships between the angles and side lengths of triangles, and it extends those relationships to circles, waves and anything else that repeats. The name comes from the Greek trigōnon (triangle) and metron (measure), so trigonometry literally means "triangle measurement." This guide covers the six trigonometric ratios, a table of standard angle values, the core formulas and identities, worked examples and the real-world applications that make the subject worth learning.

What Is Trigonometry in Simple Words?

In a right triangle, once you know one acute angle, the ratios between the sides are fixed no matter how large the triangle is. A 30° angle in a tiny triangle and a 30° angle in a huge one produce the same side ratios. Trigonometry gives those ratios names (sine, cosine, tangent and their three reciprocals) so you can find unknown sides and angles from limited information. Through the unit circle, the same functions are then defined for any angle, including angles over 90° and negative angles. That step is what lets trigonometry describe rotation, sound, light and alternating current.

Trigonometry vs. Geometry

Geometry studies the properties of shapes and space in general: points, lines, polygons, circles and solids in two or three dimensions. Trigonometry grew out of geometry but focuses on one question, which is how angles relate to lengths. It then goes beyond classical geometry by treating sine and cosine as functions with graphs, periods and identities, which is why trigonometry is the bridge to precalculus and calculus.

A Brief History of Trigonometry

Ancient Egyptian and Babylonian mathematicians worked with side ratios for building and astronomy. The Greek astronomer Hipparchus (2nd century BCE) is often called the founder of trigonometry because he compiled one of the first tables of chords, and Ptolemy extended that work in his Almagest in the 2nd century CE. Indian mathematicians, including Aryabhata around 500 CE, replaced chords with half-chords, the ancestor of the modern sine. Scholars in the medieval Islamic world developed the remaining functions and spherical trigonometry for astronomy and for finding the direction of Mecca, and their work later shaped trigonometry in Europe.

The Parts of a Right Triangle

Every right-triangle ratio is defined relative to one acute angle, usually written θ (theta):

• Hypotenuse: the longest side, opposite the 90° angle.
• Opposite side: the side across from θ. Some textbooks call it the perpendicular.
• Adjacent side: the side next to θ that is not the hypotenuse. Some textbooks call it the base.

The opposite and adjacent sides swap when you switch to the other acute angle, which causes a lot of mistakes. Always label the sides from the angle you are working with.

The Six Trigonometric Ratios

The memory aid SOH-CAH-TOA covers the three main ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse and Tangent = Opposite/Adjacent. The other three are reciprocals.

Inverse functions work the other way: when you know a ratio and need the angle, use θ = sin⁻¹(opposite/hypotenuse), θ = cos⁻¹(adjacent/hypotenuse) or θ = tan⁻¹(opposite/adjacent). Note that sin⁻¹ means inverse sine (arcsin), not 1/sin.

Trigonometry Table of Standard Angles

These exact values come up constantly in homework and exams. "Undefined" means the ratio would require dividing by zero.

Some textbooks write √3/3 as 1/√3 and √2/2 as 1/√2; the values are equal. For the angles that lie on the axes:

Memory trick: for 0°, 30°, 45°, 60° and 90°, sine follows the pattern √0/2, √1/2, √2/2, √3/2, √4/2, which simplifies to 0, 1/2, √2/2, √3/2 and 1. Cosine is the same list in reverse, and tangent is sine divided by cosine.

The Unit Circle

The unit circle is a circle of radius 1 centered at the origin. Draw an angle θ from the positive x-axis, and the point where it meets the circle has coordinates (cos θ, sin θ). This definition works for every angle, not just those inside a right triangle, and it shows why sine and cosine always stay between −1 and 1.

• Radians: one full turn is 360° or 2π radians, so 180° = π radians. To convert degrees to radians, multiply by π/180.
• Signs by quadrant: all six functions are positive in Quadrant I, sine and cosecant in Quadrant II, tangent and cotangent in Quadrant III, and cosine and secant in Quadrant IV. The mnemonic "All Students Take Calculus" helps.
• Reference angles: an angle's trig values match those of its reference angle (the acute angle it makes with the x-axis), with the sign set by the quadrant. For example, sin 150° = sin 30° = 1/2, while cos 150° = −cos 30° = −√3/2.

Key Trigonometry Formulas and Identities

An identity is an equation that holds for every angle where both sides are defined. These are the ones you will use most.

Reciprocal and Quotient Identities

• csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
• tan θ = sin θ / cos θ, cot θ = cos θ / sin θ

Pythagorean Identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

Here sin²θ means (sin θ)².

Even-Odd and Cofunction Identities

• sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ
• sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ

Sum, Difference and Double-Angle Formulas

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B
• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
• tan 2θ = 2 tan θ / (1 − tan²θ)

Half-Angle Formulas

• sin(θ/2) = ±√[(1 − cos θ)/2]
• cos(θ/2) = ±√[(1 + cos θ)/2]

The sign depends on the quadrant in which θ/2 lies.

Laws for Any Triangle

For a triangle with sides a, b and c opposite angles A, B and C:

• Law of sines: a/sin A = b/sin B = c/sin C
• Law of cosines: c² = a² + b² − 2ab cos C, and likewise for a² and b²
• Area: Area = ½ab sin C

Use the law of sines when you know two angles and a side (or two sides and a non-included angle, where you should check for the ambiguous case). Use the law of cosines when you know two sides and the included angle, or all three sides.

Trigonometry Examples With Solutions

Example 1: Height of a Tree

You stand 20 m from the base of a tree and measure a 35° angle of elevation to the top. The height h is opposite the angle and your distance is adjacent, so tan 35° = h/20. That gives h = 20 × tan 35° ≈ 20 × 0.700 ≈ 14.0 m. Add your eye height if you measured from eye level.

Example 2: Missing Side From the Hypotenuse

A right triangle has a 45° angle and a hypotenuse of 10 units. The side x opposite the 45° angle satisfies sin 45° = x/10, so x = 10 × √2/2 = 5√2 ≈ 7.07 units.

Example 3: A Ladder Against a Wall

The foot of a ladder is 5 m from a wall, and the ladder makes a 60° angle with the ground. The 5 m distance is adjacent to the angle and the ladder is the hypotenuse, so cos 60° = 5/L. Since cos 60° = 1/2, the ladder is L = 10 m long. The height it reaches on the wall is 5 × tan 60° = 5√3 ≈ 8.66 m. Mixing up the ladder length and the wall height is a classic error, so label the triangle before choosing a ratio.

Example 4: Double-Angle Formula

If sin x = 0.6 and x is acute, then cos x = √(1 − 0.6²) = √0.64 = 0.8. So sin 2x = 2(0.6)(0.8) = 0.96. If x were in Quadrant II, cos x would be −0.8 and sin 2x would be −0.96.

Example 5: Solving a Trigonometric Equation

Solve 2 sin x − 1 = 0 for 0° ≤ x < 360°. Rearranging gives sin x = 1/2. Sine is positive in Quadrants I and II and the reference angle is 30°, so x = 30° or x = 150°.

Example 6: Law of Sines

A triangle has A = 50°, B = 70° and side b = 8 opposite angle B. Then a = 8 × sin 50° / sin 70° ≈ 8 × 0.7660 / 0.9397 ≈ 6.52. The third angle is C = 180° − 50° − 70° = 60°.

Example 7: Simplifying an Expression

Simplify 2sin²x − 2cos²x. Replacing sin²x with 1 − cos²x gives 2 − 4cos²x. By the double-angle formula, the same expression also equals −2cos 2x.

Real-World Applications of Trigonometry

• Surveying and construction: surveyors find distances and heights they cannot measure directly, and builders use angles for roof pitch, stairs and ramps. A wheelchair ramp at the ADA's maximum running slope of 1:12 rises at about 4.8°, since tan⁻¹(1/12) ≈ 4.76°.
• Navigation and aviation: pilots and ship navigators use bearings and triangles to plot courses and correct for wind or current.
• Astronomy: astronomers find the distance to nearby stars with parallax, the small apparent shift in a star's position as Earth orbits the Sun.
• Physics and engineering: engineers split forces into horizontal and vertical components with sine and cosine, and model sound, vibrations and alternating current as sine waves.
• Computer graphics and robotics: rotating an object on screen or working out where a robot arm will reach relies on sine and cosine.
• Forensic science: analysts estimate the angle at which a blood drop struck a surface from the stain's shape, using sin θ = width/length.
• Signal processing: Fourier analysis breaks complex signals into sums of sine and cosine waves, which underpins audio and image compression.

Tips for Learning Trigonometry and Mistakes to Avoid

• Check your calculator mode. Degree versus radian mode causes more wrong answers than anything else. In radian mode, sin 30 is about −0.988, not 0.5.
• Memorize the standard angles. Use the √n/2 pattern above, or sketch the 30-60-90 and 45-45-90 triangles.
• Draw and label first. Mark the angle, then label opposite, adjacent and hypotenuse before choosing a ratio.
• Use reference angles for angles beyond 90°, then apply the correct quadrant sign.
• Round at the end. Keep full calculator precision until the final step.
• Watch for undefined values. The tangent and secant of 90° do not exist because cos 90° = 0.
• Practice identities by proving them. Working one side into the other builds the pattern recognition you need on exams and later in calculus.

If your homework runs on an online system, our guide on how to solve WeBWorK problems explains how to enter trig functions correctly, and these math test preparation strategies help when an exam is coming up.

Getting Help With Trigonometry

Start with your instructor's office hours or your campus tutoring center when a topic will not click. For one-on-one support, our precalculus homework help and math homework help services connect you with tutors: you post your problem set with instructions and a deadline, compare tutor bids and profiles, and message your tutor about each step. Use worked solutions to check and understand your own work, and follow your course's academic integrity rules. When you are ready, post your trigonometry question.

Frequently Asked Questions

What Is Trigonometry in Simple Terms?

Trigonometry is the part of math that connects angles to lengths. In a right triangle, each angle fixes the ratios between the sides, and trigonometry names those ratios sine, cosine and tangent. Knowing one angle and one side lets you find the rest. Using the unit circle, the same functions describe any angle, so trigonometry also models rotation, waves and other repeating patterns.

What Are the Three Main Trigonometric Functions?

The three main functions are sine, cosine and tangent. In a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent, which the mnemonic SOH-CAH-TOA summarizes. Their reciprocals, cosecant, secant and cotangent, make up the full set of six trigonometric functions you will see in precalculus and calculus.

Who Is Known as the Father of Trigonometry?

The Greek astronomer Hipparchus, who lived in the 2nd century BCE, is usually called the father of trigonometry because he compiled one of the earliest known tables of chords for astronomical calculations. Trigonometry developed over many centuries, though. Ptolemy, Indian mathematicians such as Aryabhata and scholars in the medieval Islamic world all made major contributions to the subject as it is taught today.

What Is Trigonometry Used for in Real Life?

Trigonometry is used to find heights and distances in surveying and construction, plot courses in navigation and aviation, and measure distances to nearby stars through parallax. Engineers and physicists use it to break forces into components and to model sound, light and alternating current as waves. It also drives computer graphics, robotics and the signal processing behind audio and image compression.

How Do You Memorize the Trigonometry Table?

For 0°, 30°, 45°, 60° and 90°, write sine as √0/2, √1/2, √2/2, √3/2 and √4/2, then simplify. Cosine is the same list in reverse order. Tangent is sine divided by cosine, and the remaining three functions are reciprocals. Sketching a 30-60-90 triangle with sides 1, √3 and 2 and a 45-45-90 triangle with sides 1, 1 and √2 also lets you rebuild the values quickly.

What Is the Difference Between Degrees and Radians?

Degrees and radians are two units for measuring angles. A full circle is 360 degrees or 2π radians, so 180° equals π radians. To convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π. Calculus and many online homework systems use radians by default, so always check which unit a problem expects and set your calculator to match.

Is Trigonometry Hard to Learn?

Trigonometry feels hard mainly because it introduces many new terms and identities at once. It becomes much easier when you master right-triangle ratios first, memorize the standard angle values, and then learn the unit circle. Regular practice with mixed problems, drawing diagrams and checking calculator mode prevents most errors. Asking for help early, before gaps build up, also makes a big difference.