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MATH: TRIGONOMETRY Section V DePaul Math Placement Test

[object Object],Most of the trigonometry on the math placement test addresses the different parts of a right triangle and the relationships among these different parts. The three basic trigonometric functions— sine, cosine, and tangent —are the tools that define these relationships. Given the measure of one of the non-right angles in a right triangle, you can use these tools of trigonometry to find the characteristics of the triangle. If you are given the measure of one of the non-right angles and one of the sides, you can find all the values of the right triangle. Basic Functions and the Right Triangle If you know the measure of one of the non-right angles in a right triangle, the trigonometric functions tell you the ratio of the lengths of any two sides of the triangle. In the right triangle below, one acute angle is labeled and the sides of the triangle are labeled hypotenuse, opposite, and adjacent, according to their position relative to the angle of measure .

[object Object],If you know the measure of one of the non-right angles in a right triangle, the trigonometric functions tell you the ratio of the lengths of any two sides of the triangle. In the right triangle below, one acute angle is labeled and the sides of the triangle are labeled hypotenuse, opposite, and adjacent, according to their position relative to the angle of measure Sine: The sin of an angle is the ratio of the side opposite the angle to the hypotenuse. sin =opposite/hypotenuse Cosine: The cos of an angle is the ratio of the side adjacent the angle to the hypotenuse. cos= adjacent/hypotenuse Tangent: The tan of an angle is the ratio of the side opposite the angle to the side adjacent to the angle. tan= opposite/adjacent

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[object Object], Angles Larger than 90 0 and the Basic Functions Angles in a right triangle can never be larger than 90º, since the sum of all three angles must equal 180 0 . But in the math placement test, you may occasionally run into angles that are larger than 90 0 . It is often more intuitive to think of these in terms of the coordinate plane rather than in terms of a triangle. Angles and quadrants Below are pictured four angles in the coordinate plane. The first is the acute angle we’ve already covered in this chapter; the next three are all larger than 90 0 . The four quadrants of the coordinate plane become very important when dealing with angles that are larger than 90 0 . Each angle larger than 90 0 can be “simplified” by looking at it in the context of its own quadrant.

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[object Object], You should know the proper sign for each quadrant in an indirect way, meaning that it’s unlikely that you’ll have to do any heavy calculating when dealing with this topic. Instead, you might find a question such as: Q. If the value of sin – is .5 , what is the value of sin ? A. The first thing you should notice is that - and have the same magnitude, even if they have different signs. This means that the magnitude of sine for – and will be the same. Immediately you should understand that sin must equal either .5 or –.5. To figure out which of these values is right, you have to decide what quadrant angle resides in. Based on the graph of the sine function or from the above chart, you can see that the sine function has a positive value in quadrants I and II, and negative values in quadrants III and IV. Since sin– is equal to a positive number, .5, you know that – must represent an angle in quadrant I or II. Since angle is simply the reflection of – across the x -axis, you can see that angle must be in either quadrant III or IV. The value of sin must be negative: – . 5 is the right answer.

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[object Object], Here is another Example: Q. Given: Two Sides The right triangle below has a leg of length 5 and a hypotenuse of length 8. First, use the Pythagorean theorem to find the length of the third side: 2 b= = 6.2 Next, use trigonometric functions to solve for the acute angles: sin A = 5/8 cos B = 5/8 Now you know that sin A = 5 ⁄ 8 , but you are trying to find out the value of L A, not sin A . To do this, you need to use some standard algebra and isolate L A. In other words, you have to find the inverse sine of both sides of the equation sin A = 5 ⁄ 8 . Carrying out this operation will tell you exactly which angle between 0 o and 90 o has a sine of 5 ⁄ 8 . sin -1 (sin A) = sin -1 (5/8) L A=38.7 o You can solve for L B by using the cos –1 button and following the same steps.(51.3 o ).

[object Object], A trigonometric identity is an equation involving trigonometric functions that holds true for all angles. These identities are commonly called Pythagorean identities, because they come from the Pythagorean theorem. tan = Sin / Cos sin 2 + cos 2 = 1 You will have to by asking you to simplify a complex expression. Answering these questions has more to do with memorizing the identities and being good with algebraic substitution than it does with the theoretical concepts of trigonometry. For example: What is (cos tan ) / (sin – cos 2 )? To solve a problem like this, use the trigonometric identities to simplify the trigonometric into sines and cosines. After you have simplified the expression using the identities, it is quite likely that the expressions will simplify further due to the canceling of terms. The simplification of the expression in the example question proceeds as follows: (Cos X tan )/Sin – Cos 2 = [(Cos X Sin ) /Sin X Cos ] - Cos 2 = 1-Cos 2 = Sin 2 Simplifying the mess given to you by the problem, you get sin 2 .

[object Object], The unit circle is a circle whose center is the origin and whose radius is 1. It is defined by equation x 2 + y 2 = 1. The most useful and interesting property of the unit circle is that the coordinates of a given point on the circle can be found using only the measure of the angle. Any radius of the unit circle is the hypotenuse of a right triangle that has a (horizontal) leg of length cos and a (vertical) leg of length sin . The angle is defined as the radius measured in standard position. These relationships are easy to see using the trigonometric functions: sin = y/1 =y cos = x/1 =x tan = y/x As you can see, because the radius of the unit circle is 1, the trigonometric functions sine and cosine are simplified: sin = y and cos =x . This means that another way to write the coordinates of a point (x, y) on the unit circle is (cos , sin ), where he measure of the angle in standard position whose terminal side contains the point.

[object Object], The Unit Circle and Important Angles Using the unit circle makes it easy to find the values of trigonometric functions at quadrantal angles. For example , a 90 o rotation from the positive x -axis puts you on the positive y -axis, which intersects the unit circle at the point (0, 1). From this, you know that (cos 90 o , sin 90 o ) = (0, 1). Here is a graph of the values of all three trigonometric functions at each quadrantal angle: There are a few other common angles besides the quadrantal angles whose trigonometric Function values you should already know. Listed on the bottom-are the values of sine, cosine, and tangent taken at 30 o , 45 o and 60 o .You might recognize some of these values from the section on special triangles.

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[object Object],To convert from degrees to radians : multiply the degree measure by π / 180 . For example, 60º is equal to 60π / 180 = π / 3 radians. To convert from radians to degrees : multiply the measure in radians by 180 / π . For example, π / 4 radians is equal to 180π / 4π = 45 o . Here are the most important angle measures in degrees and radians: ---- 30 0 = π / 6 ---- 45 0 = π / 4 ---- 60 0 = π / 3 ---- 90 0 = π / 2 ---- 120 0 = 2π / 3 ---- 135 0 = 3π / 4 ---- 150 0 = 5π / 6 ---- 180 0 = c ---- 210 0 = 7π / 6 ---- 225 0 = 5π / 4 ---- 240 0 = 4π / 3 ---- 270 o = 3π / 2 ---- 300 o = 5π / 3 ---- 315 o = 7π / 4 ---- 330 o = 11π / 6 ---- 360 o = 2π

[object Object],Complementary angle identities Two angles whose sum is π/2 radians ( 90 degrees) are complementary . In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change θ to π/2 − θ, obtaining: Now, test your knowledge of the topics discussed by clicking on the sample test links given below.

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