Presentation on theme: "Trigonometric Functions. Trigonometric Identities The following identities need to be memorized: Standard/Reciprocal Identities Pythagorean Identities."— Presentation transcript:
Trigonometric Identities The following identities need to be memorized: Standard/Reciprocal Identities Pythagorean Identities Double-Angle Identities
Measure of an Angle 1 1 Terminal Side The measure (counter clockwise) of an angle is determined by the amount of rotation from the initial side to the terminal side. Initial Side
Reference Angle 1 1 Terminal Side The measure of a reference angle is the acute angle formed by the terminal side. and the x-axis. Initial Side
Coordinates on the Unit Circle 1 1 Ө (cosӨ,sinӨ) Example: Find the coordinates for a 210°angle in the unit circle. This helps us find exact values for sine and cosine.
Exact Coordinates on the Unit Circle ° 45° 60° 150° 135° 120° 210° 225° 240° 330° 315° 300° 0°0° 180° 90° 270° These coordinates tell you the exact values of cosine and sine for 16 angles. They need to be memorized.
Exact Coordinates on the Unit Circle 1 1 π / 6 π / 4 π / 3 π / 2 2π / 3 3π / 4 5π / 6 7π / 6 5π / 4 4π / 3 3π / 2 5π / 3 7π / 4 11π / 6 0 π A majority of your work in calculus will be with radians, not degrees. NOTICE: Angles with the same reference angle have the similar coordinates. This needs to be memorized.
Tricks of the Trade WARNING: Although shown, use these with great caution.
Trick #1: Special Right Triangles 1 30° 1 or π / 6 60°or π / 3 45 ° or π / 4 Use Special Right triangles to remember the first quadrant. 30°- 60°- 90° 45°-45°-90° OR Isosceles Right 45 ° or π / 4
Trick #2: Finding a Reference Angle in Radians On the left are 3 reference angles that we know exact trig values for. To find the reference angle for angles not in the 1 st quadrant (the angles at right), ignore the integer in the numerator. Bonus: Multiply the number in the numerator by the degree to find the angle’s quadrant.
Example Find the reference angle and quadrant of the following: Or 45º
Trick #3: Stewart’s Table for the First Quadrant R.A.SinCosTan Find the reference angle’s value, then the quadrant to figure out the sign (+/-) Square root # goes up 1 Reverse sine
Trick #4: Deciding whether a Trigonometric Function is Positive 1 1 S A T C All Just Sine (and cosecant) Just Tangent (and cotangent) Just Cosine (and secant) ALLSTUDENTS TAKECALCULUS
Example 1 Find the exact value of the following: Thought process The only thing required for a correct answer (unless the question says explain)
Example 2 Solve: Isolate the Trig Function What angles make the sine function 0.5? when What other angles make the sine function 0.5? Remember: There are typically 2 answers. When does 4x equal our two angles? Sine is cyclic, how often does it repeat?