Section Notes Page Trigonometric Functions; Unit Circle Approach A unit circle is a circle centered at the origin with a radius of Its equation is x y = as shown in the drawing below Here the letter t represents an angle measure The point P=(x, y) represents a point on the unit circle The following definitions are given based on this picture sin t = y csc t = y cos t = x sec t = x y x tan t = cot t = x y If we start with x y = and put in our definitions above we will have cos θ sin θ = EXAMPLE: Suppose a point on the unit circle is, Find all six trigonometric values We want to use our definitions to answer the questions We know that x = and y = because of the point given This automatically tells us that sin t = and cost = We can plug in x and y into the other equations to find the remaining trig values csct = = sect = = tan t = = = cot t = = = EXAMPLE: Use a calculator to find the approximate value of cos 4 rounded to the two decimal places Make sure your calculator is in degree mode, and then enter it into the calculator You should get 097 EXAMPLE: Use a calculator to find the approximate value of sin rounded to the two decimal places 8 You need to have your calculator in radians for this one, because of the You should get 08
Now let s look at the angle of t = 4 or At this angle, we end up with the following triangle: 4 Section Notes Page 4 4 90 Triangle 4 We can use our definitions of sine, cosine, and tangent to find exact values: 4 sin 4 = cos 4 = tan 4 = = From our above definitions we can also find the following: csc 4 = = = =, sec 4 = = = =, cot 4 = = tan cot 4 4 We just need to plug in the values of the trig functions we found above: = Now let s look at two other special angles on the unit circle We will consider t = 0 and 0 0 90 Triangle ( t = 0 ) 0 0 90 Triangle ( t = 0 ) t = 0 0 0 / / 0 / 0 / We can use our definitions of sine, cosine, and tangent again to find exact values with 0 and 0 degrees sin 0 = cos 0 = tan 4 = = = = sin 0 = cos 0 = tan 4 = = =
Section Notes Page Table of trigonometric values θ (degrees) θ (radians) sin θ cos θ tan θ 0 0 0 0 0 4 0 90 4 0 undefined cos 0 sin 0 This can be rewritten as: ( ) cos0 Now square everything inside the parenthesis: subtracting we get sin 0 Now substitute values by using the table 4 Reduce this fraction: After We can put in values right away from the table: over the bottom fraction and multiply to get: cos0 sin 0 We can subtract on the top to get: which is = We can flip cos sec cot We don t have a cotangent or secant on our table, so we will need to first convert this into a sine by using the identity: sec = and cot = From the table, we know that cos = and tan = cos( ) tan( ) We can put in these values into our expression: = = =
Section Notes Page 4 If you refer back to the previous table of trig values, you will notice that you see the same value repeated This leads to the following identities: Complementary Angle Theorem sin θ = cos( 90 θ ) csc θ = sec( 90 θ ) cos θ = sin( 90 θ ) sec θ = csc( 90 θ ) tan θ = cot( 90 θ ) cot θ = tan( 90 θ ) EXAMPLE: Write the following as an equivalent cosine expression: sin We can use a Complementary Angle Theorem for this We will use θ = cos( 90 θ ) Substituting we get sin = cos( 90 ) So our answer is: cos7 calculator you should get the same decimal EXAMPLE: Simplify and find the EXACT value: tan sec cos sin Here θ = sin = If you put both of these in your We need to use a Complementary Angle Theorem to get rid of the angle The only formula we can use for secant is sec θ = csc( 90 θ ) We will get: sec = csc( 90 ) We will get: sec = csc Now our problem becomes: tan csc cos Let s now change everything into sines and cosines: sin cos We can cross cancel the sines Then the cosines also cancel and we get cos sin Reference Angle an angle between 0 and 90 that is formed by the terminal side of an angle and the x-axis The reference angle is labeled below It is indicated by the double curved lines Notice that no matter where the angle is drawn it is measured from the x-axis Under each drawing it tells you how to find the reference angle: If 90 < θ < 80 then If 80 < θ < 70 then If 70 < θ < 0 then Ref angle = 80 θ Ref angle = θ 80 Ref angle = 0 θ If < θ < then If < θ < then If < θ < then Ref angle = θ Ref angle = θ Ref angle = θ
EXAMPLE: Find the reference angle for 70 Section Notes Page Since 90 < 80 < θ, we will use the formula θ 80, so the reference angle is 80 70 = 0 EXAMPLE: Find the reference angle for 9 Since < θ <, we will use the formula θ, so the reference angle is 9 0 9 = = EXAMPLE: Draw 0 in standard position and then find its reference angle First we will draw it in standard position The reference angle is indicated by the double curved lines: To find the reference angle, we use the formula above, which says that the reference angle is 80 θ So we our reference angle is: 80 0 = 0 EXAMPLE: Draw in standard position and then find its reference angle 80 We can change into degrees so we know how to graph it: = 0 standard position The reference angle is indicated by the double curved lines Now we will draw it in To find the reference angle, we use the formula above, which says that the reference angle is 0 θ So we our reference angle is: 0 0 = 0 We need to change this back into radians since the problem was originally given in radians Our reference angle is:
Sign values of sine, cosine, and tangent in each quadrant Section Notes Page Depending on which quadrant you are in the sine, cosine, and tangent functions will be either positive or negative You will need this for using reference angles to find trigonometric values The quadrants are number from to 4 counterclockwise starting with the upper right quadrant Each quadrant has a certain angle value: In quadrant : 0 < θ < 90, in quadrant : 90 < θ < 80, in quadrant : 80 < θ < 70, and in quadrant 4: 70 < θ < 0 An easy way to remember the sign chart is the phrase All Students Take Calculus The first letter of each word in the phrase tells you what is positive in each quadrant, starting in quad and going counterclockwise ALL Means all of them are positive in the first quadrant S Means sine is the only one positive in quad T Means tangent is the only one positive in quad C Means cosine is the only one positive in quad 4 How to find the trigonometric value for any angle: ) Find the reference angle ) Apply the trig function to the reference angle ) Apply the appropriate sign EXAMPLE: Find the exact value of indicate the reference angle We will follow the three steps from above cos using reference angles Draw the angle in standard position and ) First we will draw this angle in standard position The reference angle is indicated by the double curved lines We found the reference angle by taking 80 = 4 ) We need to apply the trig function to our reference angle, so we will do cos 4 = ) We need to apply the appropriate sign This is where we will use the sign chart from the last page This angle is in the second quadrant, so cosine needs to be negative here So now we can write our answer: cos =
EXAMPLE: Find the exact value of indicate the reference angle Section Notes Page 7 4 sin using reference angles Draw the angle in standard position and We can change this into degrees to see what quadrant we are in: 4 80 = 40 ) First we will draw this angle in standard position The reference angle is indicated by the double curved lines We found the reference angle by taking 40 80 = 0 This is equivalent to ) We need to apply the trig function to our reference angle, so we will do sin = ) We need to apply the appropriate sign This angle is in the third quadrant, so sine needs to be negative here So now we can write 4 our answer: sin = EXAMPLE: Find the exact value of indicate the reference angle cos0 using reference angles Draw the angle in standard position and ) First we will draw this angle in standard position The reference angle is indicated by the double curved lines We found the reference angle by taking 0 0 = 0 ) We need to apply the trig function to our reference angle, so we will do cos 0 = ) We need to apply the appropriate sign This angle is in the fourth quadrant, so cosine needs to be positive here So now we can write our answer: cos 0 =