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Tech Math 2 Angle Precision 1 0.1 or 10 0.01 or 1 Test #01 Review Angles and Accuracy of Trig Functions Trig Function Accuracy Example 2 sig. figs. sin17 0.29 , tan85 11 3 sig. figs. sin1.1 0.0192 , cos520' 0.996 4 sig. figs. cos 45.00 0.7071 , tan8013' 5.799 Calculating Other Trig Functions Definition Example cos 1 1 cot cot 82.3 0.135 sin tan tan 82.3 1 1 sec sec 22 1.1 cos cos 22 1 1 csc csc 72.67 1.048 sin sin 72.67 Section 4-4: The Right Triangle SOH CAH TOA: opp hyp opp tan adj 1 hyp sec cos adj sin Page 1 of 4 Pythagorean Theorem: a2 + b2 = c2 adj cos hyp 1 adj cot tan opp 1 hyp csc sin opp Tech Math 2 Test #01 Review Page 2 of 4 In solving a triangle, you are given three parts of a triangle (one of which must be the length of a side), and then are expected to calculate the other three parts. The sum of the angles of a triangle is 180, so for a right triangle (one angle 90), the two acute angles add up to the remaining 90 (mathematical lingo: they are complementary). To solve a right triangle: 1. Make a sketch of the triangle, label sides and angles consistently (a, b, and c for the legs and hypotenuse; A and B for the complementary angles), and label the given information. 2. Find a way to relate the unknown parts to the given information using a trig function (sine, cosine, or tangent) or the Pythagorean Theorem (a2 + b2 = c2). Try to use original given information to minimize rounding errors. 3. Check your work: a. Make sure the sides obey the Pythagorean Theorem. b. Make sure the angles add up to 180. c. Make sure unused trig functions give the right answers. d. Make sure that the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. Section 4.5: Applications of Right Triangles To solve applied right triangle problems: 1. Make a sketch of the situation. 2. Identify/draw right triangles on your sketch that connect given information to unknown information. 3. Solve the right triangle or triangles. Section 8.4: Applications of Radian Measure 1. A radian is a unit of angle measurement. There are 2 radians in a full circle, as opposed to 360. This gives us a conversion factor: 2 radians a. To convert from degrees to radians: 57.0 0.995 radians 360 360 b. To convert from radians to degrees: 0.200 radians 11.5 2 radians 2. Arc Length Formula: s = r. a. Note: Lines of latitude are labeled by the angle that is formed from the latitude to the center of the earth to the equator. 1 3. Area of a Sector of a Circle: A r 2 . 2 4. Angular Velocity: t Tech Math 2 Test #01 Review Page 3 of 4 5. Some units of angular velocity: radians radians radians a. , , second minute hour degrees degrees degrees b. , , second minute hour revolutions revolutions revolutions c. , (rpm) , second minute hour Section 9.2: Components of Vectors 1. Components of a vector are two vectors that, when added together, have a resultant equal to the original vector. (Usually, the components are perpendicular to each other and along the x and y axes, and are thus called the x- and y- components of a vector.) 2. Resolving a vector into components is calculating the components of a vector. 3. Steps for resolving a vector into x- and y- components: a. Place the vector A such that its angle is in standard position (i.e., measured counterclockwise from the x- axis). b. Calculate the x- and y- components using right triangle trigonometry (i.e., Ax = Acos , and Ay = Asin , note: these formulas only work if the angle is in standard position!). c. Check the components for correct sign and magnitude. Section 9.3: Vector Addition by Components 1. To add vectors by components: a. Resolve all vectors into components. b. Add all x-components to get the x-component of the resultant vector (Rx). c. Add all y-components to get the y-component of the resultant vector (Ry). d. Find the magnitude of the resultant vector R using the formula R Rx2 Ry2 . Ry , and then e. Find the angle of the resultant vector R by first using the formula tan 1 Rx using the signs of Rx and Ry to convert the angle into the correct quadrant. Section 9.4: Applications of Vectors 1. 2. 3. 4. In this section, translate the word problem into vectors, then do vector addition. Physics: sum of forces = 0 for a body at rest or moving at constant velocity. Physics: sum of forces = mass times acceleration for accelerating bodies. To find the displacement between two vectors, subtract vector components instead of adding them! Tech Math 2 Test #01 Review Page 4 of 4 Section 9.5: Oblique Triangles, the Law of Sines The Law of Sines a b c sin A sin B sin C OR sin A sin B sin C a b c For the ambiguous case, the side opposite the given angle must be less that the side adjacent to the given angle: B sin B sin 30 3.875 3.125 Angle B for this triangle is 180 - B from previous triangle. Section 9.6: The Law of Cosines The Law of Cosines a 2 b 2 c 2 2bc cos A cos A b2 c 2 a 2 2bc OR a 2 c2 b2 b c a 2ca cos B cos B 2ac 2 2 2 OR c 2 a 2 b 2 2ab cos C cos C a 2 b2 c2 2ab