Trigonometry

• How to Find the Third Point of a Right Triangle
  With only two vertices, the third point is not unique. Use the right-angle location and one more length or angle to calculate its coordinates.
• Expressions for sin(arctan x) and cos(arctan x) Without Trig Functions
  Using the principal range of arctan, sin(arctan x) = x/√(1+x²) and cos(arctan x) = 1/√(1+x²) for every real x.
• How to Find the Acute Angle Between Two Curves at Their First Intersection
  Find where two curves first intersect, calculate their tangent slopes there, and use the tangent-line formula to determine their acute angle.
• How Do You Graph a Sine Function Increased on One Side?
  There is no unique graph until “increased” is defined. Compare one-sided amplitude changes, vertical shifts, and smooth added terms.
• How to Calculate Inverse Trig Functions Without a Calculator
  Use principal ranges, exact unit-circle values, identities, and power series to evaluate inverse sine, cosine, and tangent by hand.
• How to Calculate Compound Angles in Trigonometry
  Use compound-angle identities to find sine, cosine, or tangent of sums and differences, with sign rules and a worked exact-value example.
• How to Calculate Sine and Cosine Without a Calculator
  Learn exact values for common angles and approximate other sine, cosine, and tangent values by reducing the angle and using Taylor series by hand.
• How to Compute the Sine of Huge Numbers
  For huge angles, compute sine by reducing the input modulo 2π. Learn why ordinary floating-point remainder can fail and when to use high precision.
• How to Find the Inverse Cosine Without a Calculator
  Find an inverse cosine by recognizing unit-circle values, using symmetry, or converting arccos to an arctangent and evaluating its series by hand.
• When Should Your Calculator Be in Radian or Degree Mode?
  Use degree mode for angles marked ° and radian mode for angles in radians, such as π/6. Convert mismatched inputs; inverse-trig answers use the selected mode.
• How to Write a Trigonometric Expression in Sine and Cosine
  Replace tangent, cotangent, secant, and cosecant with sine-and-cosine quotients, then simplify; the exact expression is needed for a unique answer.