6.1 1st april 2013
TRANSCRIPT
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Chapter 6: Trigonometry
6.1 Trig Ratios for angles in standard position.
In Grade 11 you learned the exact values of sine, cosine, and tangent for 30°, 45°, 60°, and 90° anglesfrom [0° , 360°]. This year we will expand onthat concept and learn a new way to identify angles.
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Positive and Negative Angles
If an angle rotates in a COUNTERclockwise direction fromthe positive x axis, it form a POSITVE ANGLE in STANDARD position.
Notice the angles can be greater than 360°.
When the angle rotates clockwise, then we have a negative angle.Again, it can be more than 360°.
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Angles that end up in the same standard position are calledCOTERMINAL ANGLES. This means the arm is in the same position. There are an INFINITE number of positive andnegative coterminal angles for any angle.
Ex) Give 2 positve and 2 negative angles that are coterminal to40°.
Ex) Write a negative angle that is coterminal to 100°.
Ex) Write a positive angle that is coterminal to 19°.
Ex) What reference angle in Quadrant 1 is in the same positionas 500°.
Ex) Write a statement for all angles coterminal to 30°.
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Ex) P(3,4) is a terminal point of an angle in standard position.Find the values of sine, cosine, and tangent.
Ex) P(1,3) is a terminal point of an angle in standard position.Find the values of sine, cosine, and tangent.
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Each of these three trig ratios, has a reciprocal.The reciprocal of sine is cosecant.The reciprocal of cosine is secant.The reciprocal of tangent is cotangent.
Therefore:
Ex) If the value of sin = 0.5, what is the value of csc ?
If the value of cot = 3, what is the value of tan ?
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Ex)
QUESTION TYPES:1) Given a point location, determine the trig ratios.
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Ex) If and is in Quadrant 2, find the values of the
other 5 trig functions and the possible values of from [0°,360°]
2) Given ONE ratio, find other ratios, and angles
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3) Given an angle, determine the TRIG RATIOS
Note: a) find the coterminal angle for 510°. Do you remember the EXACTvalues of SINE, COSINE, and TANGENT for this angle?
b) use your calculator to enter the NEGATIVE angles and find the ratiosusing it INVERSE feature.
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If we make the radius of the circle become "1" (known asthe UNIT CIRCLE), then:
Now sin =
cos =
tan =
You MUST remember this!!
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Trig ratios on the unit cirlce.
Remember that:
Therefore, from the above diagram; using x, y, and rif follows that:
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HOMEWORK: Pg. 474 #3,4,5,6 (from Grade 11), 7a), 8, 9,10.11a)b), Mult. Choice #1,2.
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Notice that the size of the circle is NOT important in trig ratios..Since the triangle are SIMILAR, the results are the same!
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