The Fundamental Identities

MA 154 Lesson 3 Delworth 6.2 Trigonometric Functions of Angles The Fundamental Identities: (1) The reciprocal identities: 1 1 1 csc  sec  cot  sin cos tan (2) The tangent and cotangent identities: sin cos tan  cot  cos sin (3) The Pythagorean identities: sin2  cos2  11 tan2  sec2 1 cot2  csc2

Take the simple right triangle with sides 3, 4 and 5 with  opposite the side of length 4.

Find sin and cos, now find sin2  cos2

sin2  cos2  1 is a Pythagorean identity since it is derived from the Pythagorean Theorem.

Divide both sides by sin2  to find another Pythagorean identity. Divide both sides by cos2  to find yet another one.

Each of the three Pythagorean identities creates two more identities by subtracting a term from the left side to the right side. sin2  cos2  1 1 tan2  sec2 1 cot2  csc2 sin2  1 cos2 tan2  sec2 1 cot2  csc2  1 cos2  1 sin2 1 sec2  tan2 1 csc2  cot2

 Verify the identity by transforming the left side into the right side. tan cot  1 sin3cot3 cos3

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  MA 154 Lesson 3 Delworth 6.2 Trigonometric Functions of Angles     sin  c o s  sec 2  2     csc  1  c o t  tan    2  sin  2 

2 2 2 (1 cos)(1 cos) sin2 cossec  1 sin

 2 2  1 sin1 tan 1 cot  tan  csc sec

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  MA 154 Lesson 3 Delworth 6.2 Trigonometric Functions of Angles Using the coordinate system,

Notice that the adjacent side corresponds to the x-value of the coordinate and the opposite side corresponds the y-value of the coordinate (x, y) The idea that the cosine of  corresponds to the x-axis and the sine of  corresponds to the y-axis is one that you need to get use to. 

If  is an angle in standard position on a rectangular coordinate system and if P(-5, 12) is on the terminal side of  , find the values of the six trigonometric functions of  .

If  is an angle in standard position on a rectangular coordinate system and if P(4, 3) is on the terminal side of  , find the values of the six trigonometric functions of  .

Find the exact values of the six trigonometric functions of  if  is in standard position and the terminal side of  is in the specified quadrant and satisfies the given condition.

III; on the line 4x – 3y = 0 II; parallel to the line 3x + y – 7 = 0

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   MA 154 Lesson 3 Delworth 6.2 Trigonometric Functions of Angles Find the quadrant containing  if the given conditions are true. a) tan  < 0 and cos  > 0 I II sin All b) sec  > 0 and tan  < 0 csc

c) csc  > 0 and cot  < 0 III IV tan cos d) cos  < 0 and csc  < 0 cot sin

Use the fundamental identities to find the values of the trigonometric functions for the given conditions:

12 tan  and cos  0 sec  4 and csc  0 5

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