CK-12 Trigonometry

by CK-12 Foundation

  Confirm Using Analytic Arguments

  The unit circle is defined as a circle whose radius is one unit, and whose center is the origin in the rectangular coordinate system. The unit circle has circumference equal to 2. By making one revolution around the unit circle the length of the arc would equal 2.

  Starting at the point , go . Also starting from the point go counterclockwise along the arc of the circle, until , that is point . In a sense, the length (which equals ) is being wrapped around the unit circle.

  If is negative, again begin at point on the unit circle and travel in the Clockwise direction to arrive at Point

  If or if , the distance traveled along the unit circle will be greater than one revolution before arriving at Point or Point .

  Following this procedure shows that for any real number , a unique point can be found on the unit circle, and the following trigonometric functions of can be defined:

  Let be a real number, and let be on the unit circle that corresponds to traveling about the unit circle as described above. Then:

  Since the unit circle helps define these trigonometric functions, these functions are usually referred to as circular functions.

  In the unit circle drawn above, for points such as , the angle that ends at that point forms the triangle shown with side lengths , and , which can be used to find the values of the six trigonometric functions. For a point in the second quadrant, such as , the marked triangle and the corresponding angle is used instead. In each quadrant, signs of the trigonometric functions change with the signs of the coordinates . For example, for , since the cosine function is defined as the value divided by the hypotenuse, or in this case, the cosine function will have a negative value in this quadrant. Similarly, for points in the third and fourth quadrants we use angles formed by the radius that meets that point and the axis, and the signs of the various trigonometric function vary accordingly to the quadrant the point is in..

  It is important to note that when and are both undefined, and when and are both undefined.

  Confirm Using Technological Tools

  When working with a calculator, be sure to know the mode it is in—are the input angles in degrees or in radians (which correspond to the length traveled around the circumference of the unit circle)? In most standard graphing calculators press the mode key and in the display there is a line that shows radian and degree. Identify the input that will be used by moving the cursor to the desired angle input and pressing enter to highlight that angle preference. This preference needs to be changed whenever the input angle changes from radian to degrees or degrees to radians.

  Reciprocal trigonometric functions such as and cot require the use of the basic trigonometric functions, and and the reciprocal key: or . For example to find the , first find the and then press the or key

  Alternative Forms

  The unit circle can help determine the sign of any trigonometric function. Consider the angle , that brings us around the unit circle. For example in the following diagram, if I brings us to we consider the angle , while if brings us to the third quadrant to points we consider the angle . At , what are the signs of the and values? Once that question is answered, the signs of the six trigonometric functions can be found. In the example where is in the third quadrant, the signs of the trigonometric functions are: is negative ( is negative), is negative ( is negative), is positive ( is positive). Since the reciprocal functions agree in sign, is negative (like ), is negative (like ) and is positive (like ).

  What would be the sign of each of the trigonometric functions in the fourth quadrant. First try to visualize the unit circle, and then ask what the signs of the and coordinates of any point in the fourth quadrant? Once those signs are known, the sign of each of the trigonometric functions are also know.

  In this quadrant, we have negative, positive, negative, and correspondingly for the co-functions.

  Reviewing all of the signs of the trigonometric functions from the first quadrant through the fourth, a short cut method is found for remember which trigonometric functions (and therefore their reciprocal functions) are positive:

  To remember which quadrants the three fundamental trigonometric functions are positive, there is a mnemonic: All Students Take Calculus.

  All functions are positive in the quadrant.

  Sine is the only primary function that is positive in the quadrant.

  Tangent is the only primary function that is positive in the quadrant.

  Cosine is the only primary function that is positive in the quadrant.

  If a trigonometric value is given, there are two possible angles, , where or in radian notation whose trigonometric value will equal the given value.

  For example, if could be a first or second quadrant angle- that is could be either or , alternatively, in radian form: or

  However, if more information is provided, such as knowing that tan is known to be negative, then there is only one possible solution, in this case, knowing that is positive, the angle must be in either the first or second quadrant, and is negative, meaning that the angle must be either in the second or fourth quadrant reveals that if both are to be valid, that the angle must be a second quadrant angle- or

  An alternative form of this problem may be: Given , find the value of .

  In this situation use the Pythagorean Trigonometric Identity: ,

  substitute to obtain: or

  or

  Notice that there are two possible solutions to this problem, since only one bit of information, , was given.

  Visualize the unit circle and utilize the mnemonic mentioned previously. Given that the sine of the angle is positive and is in the first quadrant, the visualization yields the result that a second quadrant angle will also satisfy the given information. Therefore the angle can be either a first or a second quadrant angle, and the cosine of these angles, is either positive or negative as the algebra above proved.

  Lesson Summary

  The unit circle provides a mental image of several important features of the trigonometric functions. Another name for the image created when thinking of the unit circle is the wrapping function. The length of the line segment when wrapped around the unit circle helps to visualize the coordinate that is generated on the circle. The value of each of the six trigonometric functions can be found in terms of and . The wrapping function also reveals that when the length of the line segment exceeds the circumference of the circle , the value of the functions repeat. The same is true when the length of the line segment exceed or , etc. This helps to demonstrate the PERIODIC nature of trigonometric functions.

  Review Questions

  and . Find .

  and . Find the exact values of remaining trigonometric functions.

  find the value(s) of .

  , and is a second quadrant angle. Find the exact values of remaining trigonometric functions.

  is a point on the terminal side of . Find the exact values of the six trigonometric functions.

  is a point on the terminal side of . Find the exact values of the six trigonometric functions.

  Verify using: the sides , and of a right triangle, in the first quadrant

  the ratios from a triangle

  Factor:

  Simplify using the trig identities

  Prove (the alternative form of the Trig Pythagorean Identity)

  Review Answers

  If , it must be in either Quadrant II or IV. Because , we can eliminate Quadrant IV. So, this means that the is negative. (All Students Take Calculus) From the Pythagorean Theorem, we find the hypotenuse:

  So, or (Rationalize the denominator)

  If , then , sine is negative, so is in either Quadrant III or IV. Because , we can eliminate Quadrant IV, therefore is in Quadrant III. From the Pythagorean Theorem, we can find the other leg:

  So, or

  or

  , sine is positive in Quadrants I and II. So, there can be two possible answers for the . Find the third side, using the Pythagorean Theorem:

  I
n Quadrant I,

  In Quadrant II,

  and is in Quadrant II, so from the Pythagorean Theorem

  So, and

  If the terminal side of is on means is in Quadrant IV, so cosine is the only positive function. Because the two legs are lengths and , we know that the hypotenuse is is a Pythagorean Triple (you can do the Pythagorean Theorem to verify).

  Therefore,

  If the terminal side of is on means is in Quadrant I, so sine, cosine and tangent are all positive. From the Pythagorean Theorem, the hypotenuse is:

  Therefore, and

  Using the sides , and and in the first quadrant, it doesn’t really matter which is cosine or sine. So, becomes . Simplifying, we get: , and finally .

  becomes . Simplifying we get: and .

  Factor using the difference of cubes.

  You will need to factor and use the identity.

  To prove , first use and change .

  Verifying Identities

  Working with Trigonometric Identities

  During the course, you will see complex trigonometric expressions. Often, complex trigonometric expressions can be equivalent to less complex expressions. The process for showing two trigonometric expressions to be equivalent (regardless of the value of the angle) is known as validating or proving trigonometric identities.

  There are several procedures that can be thought of when attempting to validate a trigonometric identity.

  Procedure One: Often one of the steps for proving identities is to change each term into their sine and cosine equivalents:

  Prove the identity:

  Reducing the left side of the identity to:

  Notice when working with identities, unlike equations, conversions and mathematical operations are performed only on one side of the identity. In more complex identities sometimes both sides of the identity are simplified or expanded. The thought process for establishing identities is to view each side of the identity separately, and at the end to show that both sides do in fact transform into identical mathematical statements.

  Procedure Two: Another strategy used when proving identities is to use the Trigonometric Pythagorean Theorem:

  Prove the identity:

  Procedure Three: When working with identities where there are fractions- combine using algebraic techniques for adding expressions with unlike denominators:

  Prove the identity: :

  combine the two fractions on the left side of the equation by finding the common denominator: :

  Procedure Four: If possible, factor trigonometric expressions. Actually procedure four was used in the above example:

  is factored to

  and in this situation, the factors cancel each other.

  Prove the identity:

  Technology Note

  A graphing calculator can help provide the correctness of an identity. For example looking at: , first graph , and then graph . Examining the viewing screen for each demonstrates that the results produce the same graph.

  Lesson Summary

  When verifying a trigonometric identity there are some guidelines to follow, that will usually help.

  Work on one side of the identity- usually the more complicated looking side.

  Try rewriting all given expressions in terms of sine and cosine.

  If there are fractions involved, combine them.

  After combining fractions, if the resulting fraction can be reduced, reduce it.

  The goal is to establish identicality—so as you change one side of the identity, look at the other side for a potential hint to what to do next.

  Note in all of the above validating identities, only one side of the identity was worked on. Sometimes it is necessary to work only on the left side of an identity, sometimes only on the right side of the identity.

  Review Questions

  Verify the following identities:

  Show that is true using .

  Use the trig identities to prove

  Review Answers

  Step 1: Change everything into sine and cosine

  Step 2: Give everything a common denominator, .

  Step 3: Because the denominators are all the same, we can eliminate them.

  We know this is true because it is the Trig Pythagorean Theorem

  Step 1: Pull out a

  Step 2: We know , so is also true, therefore

  This, of course, is true, we are done!

  Step 1: Change everything in to sine and cosine and find a common denominator for left hand side.

  Step 2: Working with the left side, FOIL and simplify.

  Step 1: Cross-multiply

  Step 2: Factor and simplify

  Step 1: Work with left hand side, find common denominator, FOIL and simplify, using .

  Step 2: Work with the right hand side, to hopefully end up with .

  Both sides match up, the identity is true.

  Step 1: Factor left hand side

  Step 2: Substitute for because .

  Step 1: Find a common denominator for the left hand side and change right side in terms of sine and cosine.

  Step 2: Work with left side, simplify and distribute.

  Step 1: Work with left side, change everything into terms of sine and cosine.

  Step 2: Substitute for because

  be careful, these are NOT the same!

  Step 3: Factor the denominator and cancel out like terms.

  Plug in for into the formula and simplify.

  This is true because sin

  is

  Change everything into terms of sine and cosine and simplify.

  Sum and Difference Identities for Cosine

  Recall from earlier work with functions, that functions usually do not behave as algebraic expressions. For example if does not equal . In this example or , where as or . The important thing to remember is that what is done with algebraic expressions is usually not the same for functions, although the expression and the function look somewhat alike. This is the case with trigonometric functions. might look like it should equal , but it does not.

  Difference and Sum Formulas for Cosine

  Is there a method that can be used when a given angle can be expressed as the difference of two key angles by finding the cosine of the difference of the two angles? That is, is there an expression that can be found for ?

  Let the two given angles be and where

  Begin with the unit circle and place the angles and in standard position as shown in Figure A. Point lies on the terminal side of , so its coordinates are and Point lies on the terminal side of a so its coordinates are . Place the in standard position, as shown in Figure B. The point has coordinates and the is on the terminal side of the angle , so its coordinates are .

  Triangles in figure A and Triangle in figure B are congruent. (Two sides and the included angle, , are equal). Therefore the unknown side of each triangle must also be equal. That is:

  Applying the distance formula for each of these:

  In the difference formula for cosine,

  since and

  or

  use to obtain:

  , the sum formula for cosine

  Use Cosine of Sum or Difference Identities to Verify Other Identities

  The sum/difference formulas for cosine can be used to establish other identities:

  For example: Find an equivalent form of using the cosine difference formula

  or

  or , that is (Identity A)

  This identity can be used to establish the equivalence for

  Let in equation A to obtain: or or

  That is (Identity B)

  Use Cosine of Sum or Difference Identities to Find Exact Values

  The sum and difference formulas for cosine can be used to find exact values when and are key angles:

  For example, to find the exact value of , use the difference formula where and or

  Applying the Sum and Difference Identities

  To find the , first ask what two key angles when added (or subtracted) will yield ? The
re may be more than one pair of key angles that can achieve this goal. A key angle is an angle such as , because the trigonometric values for that angle is known in fraction form. or

  , substitute known values for key angles:

  or

  Now find the value of , in fraction form only:

  , notice that and

  or

  , can be verified using a calculator in radian mode.

  Technology Note

  a. Recall that by graphing both sides of an identity such as: , using a graphing calculator can provide evidence about the correctness of the identity.

  b. For , use the calculator to first find the , then find the value of to verify that the values are identical.

  Lesson Summary

  Trigonometric functions have interesting patterns and behaviors. The important issue to remember is that what may seem obvious to students first learning about these functions may not, and usually are not, correct. An example of this would be that the does NOT equal . Another thing to observe and remember is that no matter how complicated in appearance a trigonometric identity may be, all of these identities are derived from basic geometric principles as seen when deriving the sum formulas for cosine.