The Trig Review Page The Trig Review Page
UNH Mathematics Center

  1. Fundamental Trigonometric Identities
    sin x = 1
    csc x
    		       cos x = 1
    sec x
    		       tan x = 1
    cot x
    		 = sin x
    cos x
    csc x = 1
    sin x
    		       sec x = 1
    cos x
    		       cot x = 1
    tan x
    		 = cos x
    sin x

  2. Pythagorean Identities
    sin2x + cos2x = 1       1 + tan2x = sec2x       cot2x + 1 = csc2x

  3. Negative-Angle Identities
    sin(-q) = -sin(q)       cos(-q) = cos(q)       tan(-q) = -tan(q)
    csc(-q) = -csc(q)       sec(-q) = sec(q)       cot(-q) = -cot(q)

  4. Trigonometric Identities for the Sum and Difference of Two Angles
    cos(A + B) = cos(A)cos(B) - sin(A)sin(B)     cos(A - B) = cos(A)cos(B)+sin(A)sin(B)
    sin(A + B) = sin(A)cos(B) + cos(A)sin(B)     sin(A - B) = sin(A)cos(B)-cos(A)sin(B)
    tan(A+B) = tan A + tan B
    1 - tan A tan B
    		       tan(A-B) = tan A - tan B
    1 + tan A tan B

  5. Double Angle Identities
    sin 2x = 2 sin x cos x       tan 2x = 2 tan x
    1 - tan2x
    cos 2x   =  cos2x - sin2x   =  1 - 2 sin2x   =  2 cos2x - 1

  6. Half-Angle Identities
    sin2 q = 1 - cos 2q
    2
    		      cos2 q = 1 + cos 2q
    2
    		      tan2 q = 1 - cos 2q
    1 + cos 2q

    Each half-angle formula produces the square of a trig value. You select the appropriate algebraic sign for the trig value, by knowing the appropriate quadrant for the rotation q.

  7. Trigonometric Functions on the Unit Circle

    A rotation's sine, cosine, and tangent values are easily read from its sketch on a unit circle. The point where the final-side ray meets the circle has coordinates (cos q,sin q). The slope of the final-side ray is tan q.

    The illustrated rotation, q = 2p/3, has its final-side in the second quadrant. The final-side's vertical component is sin 2p/3 = Ö3/2 @ 0.866, and its horizontal component is cos 2p/3 = -1/2. Its slope, -Ö3, is tan 2p/3.

  8. Trigonometric Ratios in Right Triangles

  9. Special Right Triangles

  10. Values of Trigonometric Functions for some Special Angles
    angle, measured in radians 0 p/6 p/4 p/3 p/2 p 3p/2 2p
    angle, measured in degrees 0 30 45 60 90 180 270 360
    angle's sine value 0 1/2 Ö2/2 @ 0.707 Ö3/2 @ 0.866 1 0 -1 0
    angle's cosine value 1 Ö3/2 @ 0.866 Ö2/2 @ 0.707 1/2 0 -1 0 1
    angle's tangent value 0 Ö3/3 @ 0.577 1 Ö3 @ 1.732 dne 0 dne 0

  11. Inverse Trigonometric Functions

    The alternative names sin-1x, cos-1x, tan-1x are possibly confusing but nevertheless widely used. In this context the -1 indicates an inverse function, not a reciprocal number: sin-1 x means arcsin x, not (sin x)-1.

File translated from TEX by TTH, version 2.72.
On 19 Feb 2001, 14:13.