Trigonometric Review Radians In this course we use radians to - PDF document

3 Trig Review P. Danziger Trigonometric Review Radians In this course we use radians to measure angles. The circumference of a circle radius r is 2 πr . So in traversing a unit circle we travel a distance 2 π . If we traverse half way we travel a distance π . Radians measure the distance around the unit cir- cle we would travel. 1

3 Trig Review P. Danziger Principal Angles Quadrant I Quadrant II Radians Degrees Radians Degrees π 90 0 0 2 π 2 π 30 120 6 3 π 45 3 π 135 4 4 π 60 5 π 150 3 6 Quadrant III Quadrant IV Radians Degrees Radians Degrees 3 π 270 180 π 2 7 π 210 5 π 300 6 3 5 π 225 5 π 315 4 4 4 π 240 11 π 330 3 6 2 π ∼ 360 o 2

3 Trig Review P. Danziger Trigonometric Functions Given a right angle triangle � � � � r � b � � � � θ � a sin θ = b r = Opposite over Hypotenuse cos θ = a r = Adjacent over Hypotenuse tan θ = b a = Opposite over Hypotenuse Note that tan θ = sin θ cos θ . OHAHOA - Oh Heck Another Hour Of Algebra (sin, cos, tan). Theorem 1 (Pythagoras’ Theorem) For any angle θ cos 2 θ + sin 2 θ = 1 3

3 Trig Review P. Danziger The sign of sin, cos and tan in other quadrants is determined by the CAST rule: S A II I t t T C III IV C - cos is positive in Quadrant IV A - All are positive in Quadrant I S - sin is positive in Quadrant II T - tan is positive in Quadrant III Otherwise sin, cos and tan are negative. � ❅ � ❅ b b � ❅ � ❅ t t a − a a − a t t � ❅ � ❅ − b ❅ − b � ❅ � 4

3 Trig Review P. Danziger Theorem 2 for any angle θ sin( − θ ) = − sin( θ ) tan( − θ ) = − tan( θ ) cos( − θ ) = cos( θ ) Note that adding 2 π to an angle yeilds effectively the same angle (once more round the circle), so this does not affect the values of trigonometric functions. So for any value of θ : sin(2 π + θ ) = sin( θ ) tan(2 π + θ ) = tan( θ ) cos(2 π + θ ) = cos( θ ) 5

3 Trig Review P. Danziger Principal Values You are expected to know the following values for trig functions. sin θ cos θ tan θ θ sin θ cos θ tan θ θ √ 1 3 1 π √ 0 0 1 0 6 2 2 3 π √ √ 1 0 − 3 1 2 π 3 3 2 2 0 − 1 0 π 3 π 1 1 π 1 1 0 √ √ − 2 4 2 2 As well as the corresponding angles in the other three quadrants. 6

3 Trig Review P. Danziger Calculating Angle with the Axis Given the coordinate values ( a, b ) we wish to find the angle made with the x − axis, θ . If a � = 0 t � If b ≥ 0 � r b � � θ θ = tan − 1 � b � � a a If b < 0 a ❅ θ θ = π + tan − 1 � b � ❅ r b a ❅ ❅ If a = 0 ❅ t If b > 0 , θ = π t 2 b If b < 0 , θ = 3 π 2 b t 7