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Math 150 Trigonometry

  • If you are considering taking a lower-level class than Math 150:
  • If you are considering taking a higher-level class than Math 150:
 
  • Sample of what you should know before taking Math 150:
    1. Given g of x equals 2 x squared minus 5 x minus 3 
    a. Find g of open parenthesis negative 3 close parenthesis
    b. Solve g of x equals 0
    2. Solve start fraction 32 over x squared minus 25 end fraction equals start fraction 4 over x plus 5 end fraction plus 2 over x minus 5 end fraction
    3. Graph the quadratic function f of x equals 2 times open parentheses x minus 3 close parentheses squared minus 1 .  Find the domain and range of the function.
    4. Find the measure of the numbered angles, given the lines m and n are parallel.
    Parallel horizontal lines labeled m on the top line and n on the bottom line with a third line that slopes upward from left to right cutting through m and n comma the top left angle formed by the third line cutting the m line is 131 degrees with the other angles labeled as 1 for the top right angle 2 for the bottom left angle and 3 for the bottom right angle comma the 4 angles formed by the third line cutting line n are labeled as 4 for the top left angle 5 for the top right angle 6 for the bottom left angle and 7 for the bottom right angle
      • Solution
        Angle 1 has measure 49 degrees since it is the supplement of 131 degrees.
        Angle 2 has measure 49 degrees since Angle 1 and Angle 2 are vertical angles.
        Angle 3 is 131 degrees since it is the supplement of Angle 2.
        Angle 4 is 131 degrees since Angle 3 and Angle 4 are alternate interior angles.
        Angle 5 is 49 degrees since it is the supplement of Angle 4.
        Angle 6 is 49 degrees since Angle 5 and Angle 6 are vertical angles.
        Angle 7 is 131 degrees since Angle 4 and Angle 7 are vertical angles.
    5. Solve for h
    A right triangle with the 90 degree angle at the top labeled at which point is labeled as capital B and the other two other points of the triangle labeled as capital A on the bottom left and capital C on the bottom right comma the side of the triangle connecting capital A and capital B is labeled as 9 comma the side of the triangle connecting capital C and capital B is labeled as 12 comma a dotted line is formed by dropping a line from capital B down to the base of the triangle which is the line connecting capital A and capital C forming a 90 degree angle with a c comma the point where the dotted line meets the line a c is labeled as capital d and the dotted line itself is labeled as lower case h
      • Solution
        First solve for side AC using the Pythagorean Theorem.
        9 squared plus 12 squared equals open parenthesis a c close parenthesis squared
        225 equals open parenthesis a c close parenthesis squared
        a c equals plus or minus 15
        Since side lengths must be positive we have a c equals 15
        Note the triangle ABC is similar to ADB. Similar triangles have corresponding sides in proportion.  Side AB of triangle ADB corresponds to side AC of triangle ABC and sides BC and CB correspond.  This leads to the proportion.
        9 fifteenths equals h over 12
        15 h equals 108
        so h equals 7 point 2
     
     
  • Sample of what you will learn in Math 150:
    1. Graph the trigonometric function over one complete period.  y equals 2 sine open parenthesis 3 x minus start fraction pi over 4 end fraction close parenthesis minus 1
      • Solution
        1. Find an interval whose length is one period.
        0 less than or equal to 3 x minus start fraction pi over 4 end fraction less than or equal to 2 pi
        Start fraction pi over 4 end fraction less than or equal to 3 x less than or equal to start fraction 9 pi over 4 end fraction
        Start fraction pi over 12 end fraction less than or equal to x less than or equal to start fraction 3 pi over 4 end fraction
        2. Divide the interval open bracket start fraction pi over 12 end fraction comma start fraction 3 pi over 4 end fraction close bracket  into four equal parts to obtain the five key x-values.
        start fraction pi over 12 end fraction comma start fraction pi over 4 end fraction comma start fraction 5 pi over 12 end fraction comma start fraction 7 pi over 12 end fraction comma start fraction 3 pi over 4 end fraction
        3. Evaluate the function at these x-values. These points will be the maximum points, minimum points and point that intersect the line y = -1.
        A table with paired x and y values as follows comma x equals start fraction pi over 12 end fraction and y equals negative 1 comma x equals start fraction pi over 4 end fraction and y equals 1 comma  x equals start fraction 5 pi over 12 end fraction and y equals negative 1 comma x equals start fraction 7 pi over 12 end fraction and y equals negative 3 comma  x equals start fraction 3 pi over 4 end fraction and y equals negative 1
        4. Plot the points found in the table and join them with a sinusoidal curve.
        a wave that starts at the point start fraction pi over 12 end fraction comma negative 1 and curves upward from left to right to the point start fraction pi over 4 comma 1 then curves downward from left to right to the point start fraction 7 pi over 12 end fraction comma negative 3 then curves upward from left to right to the point start fraction 3 pi over 4 end fraction comma negative 1
    2. Find all radian solutions of the equation sine of 3x equals 1 in the interval open bracket 0 comma 2 pi close parenthesis.
      • Solution
        sine of 3x equals 1
        From the given interval 0 less than or equal to x less than 2 pi  the corresponding interval for 3x is 0 less than or equal to 3 x less than 6 pi
        Because sine equals 1 for all angles terminating on the positive y-axis, solutions over this interval are as follows
        3 x equals start fraction pi over 2 end fraction comma start fraction 5 pi over 2 end fraction comma start fraction 9 pi over 2 end fraction
        x equals start fraction pi over 6 end fraction comma start fraction 5 pi over 6 end fraction comma start fraction 3 pi over 2 end fraction
    3.  Find the value of cosine of 2 theta, given Cosine of 2 theta equals negative 12 thirteenths
    4. Prove the identity  Start fraction cosine x over 1 plus sine x end fraction plus start fraction 1 plus sine x over cosine x end fraction equals 2 secant x
    5. Find the angle between the vectors start vector negative 5 comma 12 end vector and start vector 3 comma 2 end vector