Free Printable Worksheets for learning Trigonometry at the College level

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Trigonometry

Trigonometry is a branch of mathematics that deals with the study of triangles and their relationships with angles, sides, and trigonometric functions. Trigonometry has various real-world applications such as in navigation, engineering, physics, and architecture.

Key Concepts

Trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent
Right triangles: opposite, adjacent, and hypotenuse
Unit circle: a circle with a radius of one
Trigonometric identities: Pythagorean identity, reciprocal identities, quotient identities, and co-function identities
Law of sines and law of cosines
Radians and degrees
Trigonometric graphs
Applications of trigonometry

Definitions

Sine: The ratio of the opposite side to the hypotenuse of a right triangle.
Cosine: The ratio of the adjacent side to the hypotenuse of a right triangle.
Tangent: The ratio of the opposite side to the adjacent side of a right triangle.
Cosecant: The reciprocal of the sine function.
Secant: The reciprocal of the cosine function.
Cotangent: The reciprocal of the tangent function.
Law of sines: A relationship between the ratio of the length of a side of a triangle to the sine of the angle opposite that side.
Law of cosines: A relationship between the angle and the length of the sides of a triangle using the cosine function.

Important Information

Trigonometry is used extensively in science and engineering fields.
An understanding of trigonometry is essential for higher level mathematics and physics courses.
Trigonometry can be applied to various real-world problems such as calculating distances, angles, and trajectories.

Takeaways

Trigonometry involves the study of triangles and their relationships with angles and sides.
Trigonometry uses six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.
The laws of sines and cosines can be used to solve problems involving non-right triangles.
Trigonometry has numerous real-world applications in various fields.

Here's some sample Trigonometry vocabulary lists Sign in to generate your own vocabulary list worksheet.

Word	Definition
Trigonometry	The branch of Mathematics which deals with the study of relationships between the sides and angles of a triangle
Triangle	A three-sided polygon
Hypotenuse	The longest side of a right-angle triangle, opposite the right angle
Sine	In a right angled triangle, sine of an angle is defined as the ratio between the opposite side and the hypotenuse
Cosine	In a right angled triangle, cosine of an angle is defined as the ratio between the adjacent side and the hypotenuse
Tangent	In a right angled triangle, tangent of an angle is defined as the ratio between the opposite side and the adjacent side
SOHCAHTOA	A mnemonic used to remember the definitions and formulas of the primary trigonometric functions: `Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent`
Inverse Sine	Also known as arcsine, is the inverse function of the sine function. It returns the angle whose sine is a given number.
Inverse Cosine	Also known as arccosine, is the inverse function of the cosine function. It returns the angle whose cosine is a given number.
Inverse Tangent	Also known as arctangent, is the inverse function of the tangent function. It returns the angle whose tangent is a given number.
Periodic Function	A function that repeats its values at fixed intervals or periods
Amplitude	The amplitude of a periodic function is the maximum absolute value it attains, i.e., the distance from the axis of symmetry to the peak (or the trough)
Trigonometric Identities	An equation that is true for all values of the variables involved: `sin^2 x + cos^2 x = 1, tan x = sin x/cos x, 1 + tan^2 x = sec^2 x, 1 + cot^2 x = cosec^2 x`, etc.
Law of Sines	A law used to solve triangles involving an arbitrary angle (or “non-right” triangle) in conjunction with its opposite side
Law of Cosines	A law used to solve triangles involving information about two sides and the angle in between them, or one side and two angles
Unit Circle	A circle with a radius of 1 unit, centered at the origin of a coordinate plane. It is used to understand graphing and trigonometric functions
Radian	A unit of measurement of an angle, defined as the angle intercepted by an arc of a circle of radius 1, which is of the same length as the radius. It is equal to approximately 57.3 degrees.
Degree	A unit of measurement of an angle, equal to one three hundred sixtieth of a circle; 360 degrees make up a full circle.
Cotangent	In a right angled triangle, cotangent of an angle is defined as the ratio between the adjacent side and the opposite side
Sector	A region bounded by two radii and an arc of the circle
Ambiguous Case	A situation where the given information can result in more than one triangle or no triangles at all, while solving a triangle.

Here's some sample Trigonometry study guides Sign in to generate your own study guide worksheet.

Trigonometry Study Guide

Trigonometry is the study of relationships between the sides and angles of triangles. It is used extensively in science, engineering, and mathematics. This study guide is aimed to give you an introduction and understanding of trigonometry.

Basic Concepts

Definition of Trigonometry
Trigonometric Ratios
Pythagorean Theorem
Radian and Degree Measure
Arc Length and Angular Velocity

Trigonometric Functions

Definition of Trigonometric Functions
Sine, Cosine, and Tangent Functions
Reciprocal, Cosecant, Secant, and Cotangent Functions
Inverse Trigonometric Functions

Trigonometric Identities

Fundamental Identities
Double and Half Angle Identities
Sum and Difference Identities
Law of Sines and Cosines
Vector Addition with Trigonometry

Graphing Trigonometric Functions

Amplitude, Period, and Frequency
Sinusoidal Graphs and Phase Shifts
Graphs of Sinusoidal Functions
Transformations of Graphs

Solving Trigonometric Equations

Solving Linear and Quadratic Functions
Solving Equations with the Unit Circle
Complex Numbers and Trigonometry

Applications of Trigonometry

Finding Missing Sides and Angles of Triangles
Applied Trigonometry in Science and Engineering
Using Trigonometry to Calculate Motion
Navigation and Surveying

Tips for Solving Trigonometry Problems

Identify the problem
Visualize the problem
Use trigonometric identities
Make the problem simpler
Check your work

Remember, practice makes perfect! Therefore, work through many examples to master trigonometry. Most importantly, believe in yourself and stay positive. You got this!

Here's some sample Trigonometry practice sheets Sign in to generate your own practice sheet worksheet.

Trigonometry Practice Sheet

Problem Set 1: Angles and Basic Trig Functions

Convert the angle 240° to radians.
If sin(theta) = -0.5, find the value of cos(theta).
What is the value of tan(pi/4)?
Simplify sin^2(theta) + cos^2(theta).
If cot(theta) = 4/3, find the values of sin(theta) and cos(theta).

Problem Set 2: Trig Identities

Prove that sin(-theta) = -sin(theta).
Simplify (sin(x)-1)(sin(x)+1).
Prove that tan(theta) + cot(theta) = sec(theta)csc(theta).
Simplify cos(2x) in terms of sin(x).
Prove that sin^2(theta) - cos^2(theta) = -cos(2theta).

Problem Set 3: Trig Equations and Applications

Solve for x: 2sin(x) = sqrt(3).
If an angle of depression is 12 degrees and the distance from the observer to the object is 100 meters, find the height of the object.
Find all solutions to cos(2theta) = 0.
If the hypotenuse of a right triangle is 10 and one of the acute angles is 30 degrees, find the length of the opposite side.
A ladder 10 meters long rests against a wall. If the base of the ladder is 6 meters from the wall, what angle does the ladder make with the ground?

Problem Set 4: Advanced Trig

Prove that sin(x+y) = sin(x)cos(y) + cos(x)sin(y).
Simplify cot(arctan(x)).
Find the value of sin(pi/8) using the half-angle identity.
Find the sum of the solutions to the equation sin(2theta) = -sin(theta).
If theta is acute and sin(theta) = 3/5, find sec(theta) and cot(theta).

Note: Do not include multiple choice, true/false, or fill in the blank questions.

Trigonometry Practice Sheet

Questions

What is the definition of Trigonometry?
What is the definition of a right triangle?
What is the Pythagorean Theorem?
What is the relationship between the sides of a right triangle?
What is the formula for the area of a triangle?
What is the formula for the sine of an angle?
What is the formula for the cosine of an angle?
What is the formula for the tangent of an angle?
What is the definition of an inverse trigonometric function?
What is the definition of a cotangent?
What is the formula for the cosecant of an angle?
What is the formula for the secant of an angle?
What is the formula for the cotangent of an angle?
What is the definition of a hypotenuse?
What is the formula for the law of sines?
What is the formula for the law of cosines?
What is the formula for the area of a triangle using the law of sines?
What is the formula for the area of a triangle using the law of cosines?
What is the definition of a polar coordinate?
What is the formula for converting from rectangular coordinates to polar coordinates?
What is the formula for converting from polar coordinates to rectangular coordinates?
What is the formula for the area of a triangle using polar coordinates?
What is the definition of an oblique triangle?
What is the definition of an isosceles triangle?
What is the definition of an equilateral triangle?
What is the definition of an acute triangle?
What is the definition of an obtuse triangle?
What is the definition of an inscribed angle?
What is the definition of an angle bisector?
What is the definition of an altitude of a triangle?

Trigonometry Practice Sheet

Find the exact value of sin(105°):
Find the exact value of cos(30°):
Find the exact value of tan(45°):
Find the exact value of cot(60°):
Find the exact value of sin(270°):
Find the exact value of cos(180°):
Find the exact value of tan(90°):
Find the exact value of cot(120°):
Find the exact value of sin(-135°):
Find the exact value of cos(-45°):

Here's some sample Trigonometry quizzes Sign in to generate your own quiz worksheet.

Trigonometry Quiz

Instructions: Write the answer to each problem in the right-hand column.

Problem	Answer
What is the difference between sine and cosine?
What is the Pythagorean trigonometric identity?
What is the reciprocal of tangent?
What is the inverse of secant?
What is the derivative of sin(x)?
What is the period of sin(2x)?
What is the amplitude of cos(3x)?
What is the range of tan(x)?
What is the angle of a 3-4-5 right triangle?
What is the Law of Cosines used for?

Question	Answer
What is the sine of 30 degrees?	0.5
What is the cosine of 45 degrees?	0.707
What is the tangent of 60 degrees?	1.732
What is the cotangent of 15 degrees?	6.732
What is the secant of 75 degrees?	1.155
What is the cosecant of 90 degrees?	1
What is the inverse sine of 0.5?	30 degrees
What is the inverse cosine of 0.707?	45 degrees
What is the inverse tangent of 1.732?	60 degrees
What is the inverse cotangent of 6.732?	15 degrees

Questions	Answers
What is the definition of Trigonometry?	Trigonometry is the branch of mathematics that deals with relationships between the sides and angles of triangles and the calculations based on them.
What is the unit of measurement for angles in Trigonometry?	The unit of measurement for angles in Trigonometry is degrees.
What is the sine of an angle?	The sine of an angle is the ratio between the length of the opposite side and the length of the hypotenuse in a right triangle.
What is the cosine of an angle?	The cosine of an angle is the ratio between the length of the adjacent side and the length of the hypotenuse in a right triangle.
What is the tangent of an angle?	The tangent of an angle is the ratio between the length of the opposite side and the length of the adjacent side in a right triangle.
What is the cotangent of an angle?	The cotangent of an angle is the ratio between the length of the adjacent side and the length of the opposite side in a right triangle.
What is the secant of an angle?	The secant of an angle is the ratio between the length of the hypotenuse and the length of the adjacent side in a right triangle.
What is the cosecant of an angle?	The cosecant of an angle is the ratio between the length of the hypotenuse and the length of the opposite side in a right triangle.
What is the law of sines?	The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite to that side is constant for all sides and angles in a triangle.
What is the law of cosines?	The law of cosines states that the square of the length of a side of a triangle is equal to the sum of the squares of the lengths of the other two sides minus twice the product of those two sides and the cosine of the angle between them.