Trigonometric Functions of an Acute angle Engr. Rean Navarra

Trigonometric functions on an acute angle  Let be an acute angle in standard position in a rectangular coordinate system, and let P(x,y) be any point other than the point O on the terminal side of.  Drop a vertical line from P(x,y) to the x-axis at Q(x,0). Y r x O P(x,y) Q(x,0) X

…Definition Y r x O P(x,y) Q(x,0) X

x r y P(x,y)

Definition  Let be an acute angle in standard position in a rectangular coordinate system, and let P(x,y) be any point other than the point O on the terminal side of.  If d (O, P) =then ….

Trigonometric functions of an any angle tan and sec are undefined if x = 0 csc and cot are zero if y = 0

Notes…  The trigonometric formulas does not depend on the point P(x, y) that is chosen on the terminal of.  The fundamental identities are true for trigonometric functions of any angle.

…Notes  The domains of trigonometric functions consists of all angles for which the functions is defined (where zero denominators does not occur).

…because the denominator r > 0 for any angle.

The tangent and secant are undefined if x = 0 ( if the terminal side of the angle is on y – axis).

The cotangent and cosecant are undefined if y = 0 ( if the terminal side of the angle is on x – axis).

Coordinate Signs y ( +, + ) QIVQIII QIIQI x ( -, + ) ( -, - )(+, - )

The CAST Rule for Positive Trigo Functions y QIV QIII QIIQI x ALL C OS (& Sec) T AN (&COT) S in (&CSC)

Negative trigo. Functions QIVQIII QIIQI NONE sin, tan,csc, cot sin,csc, sec, cos cos, sec,tan, cot

Example: Finding trigonometric functions of angles a.cos 135˚ b.cos 390˚

Reference Angle  The reference angle associated with is the acute angle formed by the terminal side of and the x- axis.

Reference Angles: = = 180° - = - 180° = 360° -

Example: Find the reference Angles a. Θ = 5π/3 b. Θ= 870° Ans. a. 30° b. 20°

Evaluating Trigonometric Functions for any angle  Find the reference angle associated with the angle.  Determine the sign of the trigonometric function of by the quadrant in which lies.  The value of trigonometric function of is the same, except possibly for sign, as the value of the trigonometric function.

Example: Using the reference Angle to evaluate trigonometric functions.  Find sin 240°  Find cot 495°  Sin 16π/3  Sec (-π/4)

Example: Using the reference Angle to evaluate trigonometric functions.  If tan θ = 2/3 and θ is in Q-III, find cos θ.  If sec θ = 2 and θ is in Q-IV, find the other five trigonometric functions of θ.