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Trigonometry

Browse High School Trigonometry
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    DeMoivre's theorem.
    Volume of a tank.



Solving Trig Equations with Newton's Method [06/16/2006]
How can I solve an equation like 3sin(x) = x + 1 for 0 < x < 2pi?

Solving Trig Identities [12/31/2001]
Please can you help me prove this trig. identitity: Sin X Cot X = Cos X.

Solving Trigonometric Equations in Radians [12/14/1997]
Solve for theta in radians to 2 decimals if 0 < theta < 2 pi tan theta = -0.318.

Solving Trig Problems [08/18/1997]
Find the value of Tan [(x+1)/(x-1)]^(-1) + Tan (x)^(-1).

Special Angles [01/18/2002]
Can you help me find the solution for theta?

Spiral Inside a Hexagonal Room [09/03/2003]
Two walls meet at 120 degrees, and you have a piece of cardboard with an angle of 137.5 degrees that you want to tilt until its sides are snug against the wall. How do you find the angle of tilt?

The Square Root of i [05/25/1997]
What is the square root of i?

Square Roots of Complex Numbers [03/28/1999]
Devise at least two methods for finding the square root of (a+bi).

Squares and Triangles: a Proof [8/23/1997]
Two points lie within or on the boundary of a square. Prove that the length of at least one of the lines...

Standard Angles [08/09/2001]
Why are 0, 30, 45, 60, and 90 degrees taken as standard angles in trigonometry? Why can't we take some other angles?

Steiner-Lehmus Theorem [01/28/1999]
I have a proof about an isosceles triangle that I just can't figure out...

Strategies for Proving Trigonometric Identities [05/19/2003]
How can I prove the identity (1-sin)/(1+sin) = tan + 1/cos?

Substituting to Simplify the Integral [08/04/2003]
What is the integral of tan^3 x * secx dx?

Subtracting and Multiplying Sines of Angle Multiples [04/01/2010]
Using the trig identity for the difference of sines, Doctor Jacques helps a student prove the equivalence of two trigonometric expressions involving angles that happen to be multiples of each other.

Subtraction of Trig Functions with a Phase Shift [09/14/2004]
How do you find the difference between two trigonometric functions with the same frequency, but one is phase shifted by 120 degrees?

Summation Formulas for Trigonometric Functions [04/24/2001]
How do you find the area under the curve of a trigonometric function using the definition of a limit and not an approximation? Are there summation formulas for trigonometric functions?

Sum of Sine Series [07/17/2008]
Show that for any integer n >= 1, the sum from k = 1 to n of sin(kt) is [cos(1/2t) - cos((n + 1/2)t)]/[2 sin(1/2t)].

Sum of Sines of Arbitrary Amplitudes [01/19/2012]
A student seeks the general form for the sum of two sines of different amplitudes. To combine them into a single sinusoid, Doctor Rick invokes some trigonometric identities and basic operations.

Surface Area of a Cone [06/18/1998]
What is the formula for the surface area of a cone?

Surface Area of an Oblate Spheroid [06/02/1999]
Could you please give an example of solving for the surface area of an oblate spheroid using the inverse hyperbolic sine function?

Tan 90 [07/10/1998]
Why is tan 90 undefined?

Tan, Cos, Sin [05/21/1999]
Please tell me about tan, sin, and cos. How do you know which one to use?

Tangent [02/14/2003]
Is the trigonometric ratio tangent related to the tangent of a circle?

Tangent and Secant [2/2/1995]
I need help with the following problem: tan b + cot b = sec b csc b

Tangent Function and the Unit Circle [06/06/2002]
When demonstrating the tangent function on the unit circle, why does the picture 'flip' when the angle passes through pi/2?

Tangent of 90 Degrees [5/24/1996]
Why is the tangent of 90 degrees undefined?

Tangent to Graph of y = e^x [10/09/2003]
On the graph of y = e^x is point Q with coordinates (b,a). The tangent to Q makes an angle x with the x-axis. Why is it that tan(x) = a?

Tan pi/8 [03/16/2002]
I must find the exact value of tan pi/8.

Terminal Side [04/21/1997]
Find cos(A) if the point (2,-3) is on the terminal side of A.

Three Trig Problems [6/11/1996]
1) Find tan x if ((sin x)^2)/3 + ((cos x)^2)/7 = (-sin(2x)+1)/10 2) A = 20 deg. and B = 25 deg. Find the value of (1+tanA)(1+tanB). 3) Evaluate cos 36 - cos 72.

The Top of the Tower [07/20/2001]
Given a 300-mile-tall tower on the earth, how far away from it can you go before the top disappears below the horizon?

Train in the Rain [11/21/1996]
Given a train moving in the rain, determine the actual velocity and direction of the raindrops/the train from how raindrops appear on the train windows.

Transforming Trig Functions [05/08/2001]
Express 3 sin x + 5 cos x in terms of sin x so that the graph is the same as the first equation.

Translating Trig to Algebra; Proving Trig Identities [08/09/2002]
Write as algebraic expressions in x, free of trig or inverse trig functions: sin(arccos x), tan(arcsin x), cos(arccot x), sec(arctan(x+ 2)). Prove these identities: sec^2 x + csc^2 x = sec^2 x (csc^2 x); sin^4 x + 2sin^2 x (cos^2 x) +cos^4 x = 1.

Trekkie Trig [08/13/2013]
To find a Starship U.S.S. Enterprise Constitution Class ship's distance to another object in outer space, Doctor Jeremiah uses angles of view and stereo photogrammetry.

Triangles and Law of Sines [03/27/1998]
How can you use the Law of Sines to produce more than one triangle?

A Triangle Vertex Bisection and Its Trio of New Lengths [06/08/2012]
A trigonometry student struggles to express where the bisector of a triangle vertex intersects the side opposite it; and to describe the bisector's length in terms of the triangle's side lengths and angle measures. Doctor Peterson unpacks formulas for both, along the way invoking the Law of Cosines — and another doctor's prior work.

Triangular Garden [03/18/1997]
Find the length of a fence that runs from the right angle to the hypotenuse and separates the garden into two parts of equal perimeter.

Trig and Inverse Trig Problems [7/30/1995]
I have three trig-related questions. Would it be possible to include explanations for solutions and some general rules?

Trig Based on the Unit Square [10/22/1998]
Why are the sine and cosine graphs defined in a unit circle? Why not a unit square or any other geometric shape?

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