Essential Trigonometry

The Greek words trigonon, meaning triangle, and metria, meaning measurement, are combined to form the word trigonometry. Trigonometry was simply the study of the sides and angles of the triangle together with applications to navigation, surveying, and astronomy. Trigonometry has evolved into a very powerful analytical tool when combined with the calculus.

  1. Some vocabulary words: Vertex, initial side, terminal side, positive angle, negative angle, complementary angle, supplementary angle, angles generated by more than one revolution, reference angles, reference numbers, radian measure, degree measure.
  2. Relationships: Arc length, angle, radius, area of a sector, area of a triangle, and linear velocity.
  3. Six Trigonometric Functions for Right Triangles: To define the trigonometric functions of an angle q in standard position, construct a circle of radius r, centered at the origin, and let P (x, y) be the intersection of the terminal side of q with this circle, see Fig. III a.

    Fig. III a.Fig. III b.
  4. We make the following definition:
    5.

    sin q = y/r

    cos q = x/r

    tan q = y/x

    csc q = r/y

    sec q = r/x

    cot q = x/y


    In the special case where r = 1

    sin q = y

    cos q = x

    tan q = sin q / cos q

    csc q = 1/ sin q

    sec q =1/cos q

    cot q = cos q / sin q = 1/ tan q


    Special triangle where     q = p / 4. (See Fig. III b.)

    sin q = 1 /

    cos q = 1/

    tan q = 1

    csc q =

    sec q =

    cot q = 1


    Special triangle where     q = p / 6 (derive q = p / 3 values by swapping sin and cos) (See Fig. III b.)

    sin q = 1 / 2

    cos q = / 2

    tan q = 1 /

    csc q = 2

    sec q =2 /

    cot q =

     

  5. A trigonometric identity is an equation that is true for all angles for which both sides of the equation are defined. We need to know a few basic forms which we can easily extend by knowledge of odd /even properties of the functions. The cosine is even, cos(-q ) = cos q , while the sine is odd, sin(-q ) = -sin q .

    6. Pythagorean Identities (divide the first by cos2 q to obtain the second, by sin2 q to obtain the third)

    sin2 q + cos2 q = 1

    tan2 q + 1 = sec2 q

    1 + cot2 q = csc2 q

    Complementary Angle Identities

    sin(p /2 - q ) = cos q

    cos(p /2 - q ) = sin q

    tan(p /2 - q ) = cot q

    Supplementary Angle Identities (consider the points (x, y), (-x, y), (-x, -y), (x, -y) )

    sin(p - q ) = sin q , sin(p + q ) = -sin q

    cos(p + q ) = -cos q

    tan(p + q ) = tan q

    Addition Identities

    Subtraction Identities

    sin(a + b ) = sin a cos b + cos a sin b

    sin(a - b ) = sin a cos b - cos a sin b

    cos(a + b ) = cos a cos b - sin a sin b

    cos(a - b ) = cos a cos b + sin 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 )

    The angle between two lines:

    tan q = (m2 -m1) / (1 + m2m1)

    Double Angle Identities (let b = a in the addition identities)

    sin(2q ) = 2 sin q cos q

    cos(2q ) = cos2 q - sin2 q

    tan(2q ) = 2 tan q /(1 - tan2 q )

    Pythagorean Identity gives:

    cos(2q ) = 2cos2 q - 1

    cos(2q ) = 1 - 2sin2 q

    Half-angle Identities

    sin2(q /2 ) = (1 - cos q ) / 2

    cos2(q /2 ) = (1 + cos q ) / 2

     

  6. The Law of Cosines generalizes the Theorem of Pythagoras. If the sides of a triangle have lengths a, b, and c, and if q is the angle between the sides with lengths a and b, then
    c2 = a2 + b2 -2ab     cos q . To prove this theorem just let y = b sin q and x = b cos q and use the distance formula.

    Fig. VI a.
    c2 = (x - a)2 + (y - 0)2 = (b cos     q - a)2 + (b sin q )2 = a2 + b2 (cos2 q + sin2 q ) -2ab cos q
    = a2 + b2 - 2ab cos     q .

    The Law of Sines: In any triangle with angles     a , b , g and corresponding lengths of opposite sides a, b, c, .


  7. Inverse Trigonometric Functions Given a known value from one of the six trigonometric functions how can one recover the angle?

 

Find q if sin q =/ 2.

We begin by looking for positive angles that satisfy the equation. Because sin q is positive, the angle q must terminate in either the first or second quadrant. Our knowledge of special triangles tells us that this angle may be either p /3 or 2p /3.
Our solutions are q = p /3 + 2np , n = 0, 1, 2, … and q = 2p /3 + 2np , n = 0, 1, 2, …

To define an inverse trigonometric function one must restrict the domain of the function so that it is one-to-one. For the sine and tangent functions this is typically -p /2 to p /2. For the cosine function this is 0 to p . However for the secant function, the range is not standardized. Varberg and Varberg use 0 to p , q ¹ p /2, in order to take advantage of the relationship
sec-1x = cos-1(1/x), since cos-1 is a common calculator function.

Fig. VI b.
Now consider Fig. III a. and Fig. III b. in conjunction with Fig. VI b., our knowledge of the coordinates of the angle, lengths of the horizontal and vertical sides, and the relationships that exist between the sides and angles all congeal on the unit circle. We have three figures that can be interchangeable.

VIII. Graphs of the Trigonometric functions and their Inverses