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Free tutorials and problems on solving trigonometric equations, trigonometric identities and formulas can also be found. Java applets are used to explore, interactively, important topics in trigonometry such as graphs of the 6 trigonometric functions, inverse trigonometric functions, unit circle, angle and sine law.
Interactive Trigonometry Tutorials (Applets)
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Angle in Trigonome Trigonometry try.. Understand the definition and properties of an angle in standard position Find the coterminal angle . An analytical tutorial on how to find coterminal angles. Find The Reference Angle. Analytic Angle. Analytical al tutorial to find the reference angle to an angle. Table for the 6 trigonometric functions for special angles. A table of values of sin, cos, tan, csc, sec and cot for special angles 0, 30, 45, 60 and 90 degrees. Special Angles On Unit Circle. Special angles together with their sine and cosine displayed on a unit circle. Special Triangles Used in Trigonometry. Special triangles used to obtain all six trigonometric ratios of special angles 30, 45 and 60
Skills Needed for Mathematical Problem Solving The Use of Graphing Software in the Mathematics Classroom Understanding, Teaching and Learning Mathematics Primary Math Free Graph Paper Math Videos From Analyzemath
degrees.
Trigonometric Functions
Periods Of Trigonometric Functions . The periods of all 6 trigonometric functions are explored interactively using an applet. Properties of The Six Trigonometric Functions. The properties of the 6 trigonometric functions sin (x), cos (x), tan (x), cot (x), sec (x) and csc (x) are discussed. These include the graph, domain, range, asymptotes (if any), symmetry, x and y intercepts and maximum and minimum points. Sine Function. The sine function f(x) = a*sin(bx+c)+d is explored, interactively, using a large applet. Cosine Function. An applet helps you explore the general cosine function f(x) = a*cos(bx + c) + d. Tangent Function. The tangent function f(x) = a*tan(bx+c)+d and its properties such as graph, period, phase shift and asymptotes by changing the parameters a, b, c and d are explored interactively using an applet. Secant Function. The secant function f(x) = a*sec(bx+c)+d and its properties such as period, phase shift, asymptotes domain and range are explored using an interactive applet by changing the parameters a, b, c and d. Cosecant Function. The cosecant function f(x) = a * csc ( b x + c) + d and its period, phase shift, asymptotes, domain and range are explored using an applet. Cotangent Function. The cotangent function f(x) = a * cot ( b x + c) + d is explored along with its properties susch as period, phase shift, asymptotes, domain and range.
Graph of Sine, a*sin(bx+c), Function . Graphing and sketching sine functions of the form f (x) = a*sin (bx + c; step by step tutorial. Graphs of Basic Trigonometric Functions. The graphs and properties such as domain, range, vertical asymptotes of the 6 basic trigonometric functions: sin(x), cos(x), tan(x), cot(x), sec(x) and csc(x) are explored using an applet. Sum of Sine and Cosine Functions . An interactive tutorial to explore the sums involving sine and cosine functions such as f(x) = a*sin(bx)+ d*cos(bx).
Unit Circle in Trigonometry
Trigonometric Equations and the Unit Circle. The solutions of the trigonometric equation sin(x) = a, where a is a real number are explored using an applet. Both the graph of sin(x)and the unit circle are used to explore the solutions of this equation as a changes. Unit Circle And The Trigonometric Functions sin(x), cos(x) and tan(x) . Using the unit circle, you will be able to explore and gain deep understanding of some of the properties, such as domain, range, asymptotes (if any) of the trigonometric functions.
Inverse Trigonometric Functions
Inverse Trigonometric Functions . Inverse trigonometric functions are explored interactively using an applet. Graph, Domain and Range of Arctan function. The graph of the inverse trigonometric function arctan and its
properties are explored using an applet. Graph, Domain and Range of Arcsin function. The graph and the properties of the inverse trigonometric function arcsin are explored using an applet. Solve Inverse Trigonometric Functions Questions. Questions on inverse trigonometric functions are solved and detailed solutions are presented. Also included are exercises with answers. Find Domain and Range of Arcsine Functions Find Domain and Range of Arccosine Functions
Trigonometry Questions with Solutions Angles
Trigonometry Angle Questions With Answers. Trigonometry questions related to angles in standard position, coterminal angles, complementary and supplementary angles, as well conversion from degrees to radians and vice versa, are presented. The solutions and answers are provided. Rotation, Angular and Linear Speed Questions with Answers. Questions related to angular and linear speeds of rotating objects are presented. The solutions and answers are also provided.
Trigonometric Functions
Trigonometric Functions - Questions
With Answers. Solve trigonometry questions related to trigonometric functions. The solutions and answers are provided. Simplify Trigonometric Expressions Questions With Answers. Use trigonometric identities and formulas to simplify trigonometric expressions. Find Exact Values of Trigonometric Functions - Questions With Answers . Find exact values of trigonometric functions without using a calculator.
Analytical Trigonometry Tutorials (Problems, Examples with solutions) Solving Trigonometric Equations
Trigonometric Equations. Tutorial with detailed explanations on how to solve trigonometric equations using different methods and strategies and the properties of trigonometric functions and identities.
Trigonometry Problems
Solve Trigonometry Problems. A set of problems with detailed solutions are presented. Use Sine Functions to Model Problems. Tutorial on how to use sine functions to model problems. Given data and information about a certain situation, we model it in the form f(x) = A sin (b x + c) + D or f(x) = A cos (b x + c) + D . Solve Problems Using Trigonometric Ratios. A set of problems with detailed solutions are presented.
These problems have been designed to reinforce the use of the trigonometric ratios and Pythagora's theorem. Tutorial on Sine Functions (1)Problems. Tutorial on sine function problems. Examples with detailed solutions and explanations are included. Tutorial on Sine Functions (2)Problems. This is a tutorial on the relationship between the amplitude, the vertical shift and the maximum and minimum of the sine function.
Trigonometric Identities and Their Applications
Trigonometric Identities. A list of the basic trigonometric identities. Using Trigonometric Identities . How basic trigonometric identities are used? A tutorial with Several examples with detailed solutions are presented. Verify Trigonometric Identities . How to verify trigonometric identities? Several examples with detailed solutions are presented.
Trigonometric Formulas and Their Applications
Sum and Difference Formulas in Trigonometry. A list of the sum and difference formulas involving the sine,cosine and tangent functions used in trigonometry is presented. Use of Sum and Difference Formulas in Trigonometry. Tutorial on how to use the sum and difference formulas to solve problems in trigonometry. Product Sum and Sum Product Formulas in Trigonometry. A list of the product to sum and sum to
product formulas used in trigonometry is presented.
Polar Coordinates
Convert Polar to Rectangular Coordinates and Vice Versa . Problems, with detailed solutions, where polar coordinates are converted into rectangular coordinates and vice versa are presented.
Problems and Self Tests
Solve Trigonometric Equations . 10 problems, with their answers, on solving trigonometric equations are presented. These may be used as a self test on solving trigonometric equations. Test on Graphs Trigonometric Functions. A set of questions, with their answers, on identifying the graphs of trigonometric functions sin(x), cos(x), tan(x), sec(x), csc(x), cot(x) are presented. These may be used as a self test on the graphs of trigonometric functions.
Applications Of Trigonometry
Sine Law - Ambiguous Case - applet . The ambiguous case of the sine law, in solving triangle problems, is explored interactively using an applet. Right Triangle Problems. A tutorial on right triangles problems. Examples with detailed solutions and explanations are included.
Trigonometric Tables, Formulas and Worksheets
Trigonometric tables. Trigonometric tables of all 6 trigonometric functions, with angles in degrees and radians. Copies of these tables can be downloaded. Trigonometric Indentities and Formulas. Important definitions, identities and formulas used in trigonometry. Trigonometry Calculators. Several online trigonometry calculators and solvers in this site. Free trigonometry worksheets to download
Trigonometric Identities and Formulas Below are some of the most important definitions, identities and formulas in trigonometry.
1. Trigonometric Functions of Acute Angles sin X = opp / hyp = a / c , csc X = hyp / opp = c / a tan X = opp / adj = a / b , cot X = adj / opp = b / a cos X = adj / hyp = b / c , sec X = hyp / adj = c / b ,
2. Trigonometric Functions of Arbitrary Angles sin X = b / r , csc X = r / b tan X = b / a , cot X = a / b cos X = a / r , sec X = r / a
3. Special Triangles Special triangles may be used to find trigonometric functions of special angles: 30, 45 and 60 degress.
4. Sine and Cosine Laws in Triangles In any triangle we have: 1 - The sine law sin A / a = sin B / b = sin C / c 2 - The cosine laws 2
2
2
2
2
2
2
2
2
a = b + c - 2 b c cos A b = a + c - 2 a c cos B c = a + b - 2 a b cos C
5. Relations Between Trigonometric Functions cscX = 1 / sinX sinX = 1 / cscX secX = 1 / cosX cosX = 1 / secX tanX = 1 / cotX cotX = 1 / tanX tanX = sinX / cosX cotX = cosX / sinX
6. Pythagorean Identities 2
2
sin X + cos X = 1 2
2
2
2
1 + tan X = sec X 1 + cot X = csc X
7. Negative Angle Identities sin(-X) = - sinX , odd function csc(-X) = - cscX , odd function cos(-X) = cosX , even function sec(-X) = secX , even function tan(-X) = - tanX , odd function
cot(-X) = - cotX , odd function
8. Cofunctions Identities sin(pi/2 - X) = cosX cos(pi/2 - X) = sinX tan(pi/2 - X) = cotX cot(pi/2 - X) = tanX sec(pi/2 - X) = cscX csc(pi/2 - X) = secX
9. Addition Formulas cos(X + Y) = cosX cosY - sinX sinY cos(X - Y) = cosX cosY + sinX sinY sin(X + Y) = sinX cosY + cosX sinY sin(X - Y) = sinX cosY - cosX sinY tan(X + Y) = [ tanX + tanY ] / [ 1 - tanX tanY] tan(X - Y) = [ tanX - tanY ] / [ 1 + tanX tanY] cot(X + Y) = [ cotX cotY - 1 ] / [ cotX + cotY] cot(X - Y) = [ cotX cotY + 1 ] / [ cotX - cotY]
10.
Sum to Product Formulas
cosX + cosY = 2cos[ (X + Y) / 2 ] cos[ (X - Y) / 2 ] sinX + sinY = 2sin[ (X + Y) / 2 ] cos[ (X - Y) / 2 ]
11. Difference to Product Formulas cosX - cosY = - 2sin[ (X + Y) / 2 ] sin[ (X - Y) / 2 ] sinX - sinY = 2cos[ (X + Y) / 2 ] sin[ (X - Y) / 2 ]
12.
Product to Sum/Difference
Formulas cosX cosY = (1/2) [ cos (X - Y) + cos (X + Y) ] sinX cosY = (1/2) [ sin (X + Y) + sin (X - Y) ] cosX sinY = (1/2) [ sin (X + Y) - sin[ (X - Y) ] sinX sinY = (1/2) [ cos (X - Y) - cos (X + Y) ]
13. Difference of Squares Formulas 2
2
sin X - sin Y = sin(X + Y)sin(X - Y) 2
2
cos X - cos Y = - sin(X + Y)sin(X - Y) 2
2
cos X - sin Y = cos(X + Y)cos(X - Y)
14.
Double Angle Formulas
sin(2X) = 2 sinX cosX 2
2
cos(2X) = 1 - 2sin X = 2cos X - 1 2
tan(2X) = 2tanX / [ 1 - tan X ]
15.
Multiple Angle Formulas 3
sin(3X) = 3sinX - 4sin X 3
cos(3X) = 4cos X - 3cosX 3
sin(4X) = 4sinXcosX - 8sin XcosX 4
2
cos(4X) = 8cos X - 8cos X + 1
16.
Half Angle Formulas
sin (X/2) = + or - SQRT [ (1 - cosX) / 2 ] cos (X/2) = + or - SQRT [ (1 + cosX) / 2 ] tan (X/2) = + or - SQRT [ (1 - cosX) / (1 + cosX) ] = sinX / (1 + cosX) = (1 - cosX) / sinX
17.
Power Reducing Formulas 2
sin X = 1/2 - (1/2)cos(2X)) 2
cos X = 1/2 + (1/2)cos(2X)) 3
sin X = (3/4)sinX - (1/4)sin(3X) 3
cos X = (3/4)cosX + (1/4)cos(3X) 4
sin X = (3/8) - (1/2)cos(2X) + (1/8)cos(4X) 4
cos X = (3/8) + (1/2)cos(2X) + (1/8)cos(4X) 5
sin X = (5/8)sinX - (5/16)sin(3X) + (1/16)sin(5X) 5
cos X = (5/8)cosX + (5/16)cos(3X) + (1/16)cos(5X) 6
sin X = 5/16 - (15/32)cos(2X) + (6/32)cos(4X) - (1/32)cos(6X) 6
cos X = 5/16 + (15/32)cos(2X) + (6/32)cos(4X) + (1/32)cos(6X)
18. Trigonometric Functions Periodicity sin (X + 2Pi) = sin X , period 2Pi cos (X + 2Pi) = cos X , period 2Pi sec (X + 2Pi) = sec X , period 2Pi csc (X + 2Pi) = csc X , period 2Pi tan (X + Pi) = tan X , period Pi cot (X + Pi) = cot X , period Pi 19. Trigonometric Tables. 20. Properties of The Six Trigonometric Functions. Graph, domain, range, asymptotes (if any), symmetry, x and y intercepts and maximum and minimum points of each of the 6 trigonometric functions.
