See also the
Dr. Math FAQ:
See also the
Browse High School Trigonometry
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Volume of a tank.
- Solving the Law of Tangents for an Angle [12/18/2004]
Given the Law of Tangents, (a - b)/(a + b) = tan (A/2 - B/2) / tan
(A/2 + B/2), express angle B in terms of a, b, and A.
- 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
- 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
- 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
- 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?