Cracking the SAT Subject Test in Math 2, 2nd Edition CHAPTER 9 TRIGONOMETRY DRILL EXPLANATIONS Drill 1 Trig Functions in Right Triangles 1 2 3 4 Drill 2 Completing Triangles 1 AB = 3 38; CA = 7 25; ∠B[.]
CHAPTER 9: TRIGONOMETRY DRILL EXPLANATIONS Drill 1: Trig Functions in Right Triangles Drill 2: Completing Triangles AB = 3.38; CA = 7.25; ∠B = 65° EF = 2.52; FD = 3.92; ∠D = 40° HJ = 41.41; JK = 10.72; ∠J = 75° LM = 5.74; MN = 8.19; ∠N = 35° TR = 4.0; ∠S = 53.13°; ∠T = 36.87° YW = 13; ∠W = 22.62°; ∠Y = 67.38° Drill 3: Trigonometric Identities 10 D Use FOIL on these binomials, and you get − sin2 x Because sin2 x + cos2 x = 1, you know that 1 − sin2 x = cos2 x 16 C Express tanx as The cosine then cancels out on the top of the fraction, and you’re left with 24 A , or 1 The term (sinx)(tanx) can be expressed as or The first and second terms can then be combined: Because − sin2 x = cos2 x, this expression simplifies to cosx 38 E Break the fraction into two terms, as follows: The first term simplifies to 1, and the second term becomes easier to work with when you express the tangent in terms of the sine and cosine: The whole expression then equals 1 − cos2 x, or sin2 x 45 D First, FOIL the numerator (or note the common quadratic (a + b)(a − b) = (a2 − b2) and you get − sin2 θ which is equal to cos2 θ when you rearrange sin2 θ + cos2 θ = 1 Next, factor the denominator The denominator is the common quadratic a2 + 2ab + b2 = (a + b)2 Don’t be fooled by the exponent to the power of 4; in this case a = sin2 θ and b = cos2 θ So far, the fraction looks like the following: Because sin2 θ + cos2 θ = 1, the denominator is and the expression is equal to cos2 θ, which is (D) Drill 4: Other Trig Functions 19 E Express the function as a fraction: You can then combine the terms by changing the form of the second term: This allows you to combine the terms, like this: 23 D Express the cotangent as a fraction, as follows: The sinx then cancels out, leaving you with , or secx 24 A Express the cotangent as a fraction, and the second term can be simplified: Express both terms as fractions, and the terms can be combined: , or cscx 38 E Start by putting everything into either sin or cos You’ll notice that almost everything is over cosx, so rewrite the 1s in the denominator as : Now that all the terms in each set of parentheses are over the same denominator, you can combine the numerators within each set of parentheses: Multiply the fractions in the numerator and denominator: Divide by multiplying the numerator by the reciprocal of the denominator The cos2 x will cancel, making the multiplication straightforward: Next, use the trig identity to simplify the numerator and denominator: You know that ; because cotangent is the inverse function of tangent, Therefore, , which is (E) Another way to approach this question is to Plug In Make x = 20° Now the problem reads: Secant is You , so sec can then and tan20° = 0.364 rewrite the problem: This is your target answer You can immediately eliminate (A) and (B) (because sin and cos are always between −1 and inclusive, their product must also be in between these values) Plug 20° into each answer choice (remember that comes close ) Only (E) Drill 5: Angle Equivalencies 18 A Draw the unit circle −225° and 135° are equivalent angle measures, because they are separated by 360° Or just PITA, to see which value of x works in the equation 21 D Draw the unit circle 300° and 60° are not equivalent angles, but they have the same cosine It’s a simple matter to check with your calculator Or you could just PITA 26 B PITA and use your calculator! 30 C PITA and use your calculator! 36 D Plug In a value for θ, from the ranges in the answer choices If θ= 60°, then (sin60°)(cos60°) = 0.433, which is not less than zero So cross off any answer choices that contain 60°—(A), (B), (C), and (E) 40 E Use your calculator and Plug In the numbers in each statement Remember that cot , so cot Statements I and III are both equal to 1.192; choose (E) Drill 6: Degrees and Radians Drill 7: Non-Right Triangles a = 8.26, ∠B = 103.4°, ∠C = 34.6° Your calculator will give you ∠B = 76.6°, but you need 180° − 76.6° = 103.4° in order to have an obtuse angle with the same distance from 90° as 76.6° ∠A = 21.79°, ∠B = 120.0°, ∠C = 38.21° c = 9.44, ∠B = 57.98°, ∠C = 90.02° b = 13.418 ∠B = 125.710° ∠C = 21.290° Use Law of Sines to find Cross multiply to get sin33° = sinC, then divide both sides by Use the inverse function of sine to find the unknown angle: There is another value for ∠C, which is 158.70° (because sin21.290° = sin158.70°, as discussed in the section on the unit circle), but that would result in a triangle with more than 180°, so it can be ignored If ∠C = 21.290°, then ∠B is 180 − 33 − 21.290 = 125.710° To find side b, you can use either Law of Sines or Law of Cosines; either way, its value is 13.418 Drill 8: Polar Coordinates 39 C Draw it! The x-coordinate of the point is 6 cos , or 3 The ycoordinate is 6 sin , which is 5.196 42 B Draw it! The y-value of a point is its distance from the x-axis The y-coordinate of this point is 45 B , which equals 4.949 Draw it! In rectangular coordinates, A, B, and C have xcoordinates of This means that they are placed in a straight vertical line They define a straight line, but not a plane or space 50 C Draw it! Start by converting the polar coordinates using x = rcosθ and y = rsinθ You find the points of the triangle are (0, 2), (6.928, 4), and (−6.928, 4) Once you draw the triangle, you can see that the base is the distance between (6.928, 4) and (−6.928, 4), or 13.856 The height is the distance from (0, 2) to the perpendicular base on the line y = 4, so h = 2 The area of the triangle is ; therefore, , which is (C) Comprehensive Trigonometry Drill D Because and intersect at point C, ∠ACB ≅ ∠ECD Because both triangle ABC and triangle CDE are right triangles, ∠ABC ≅ ∠EDC Therefore, sinθ = sine of ∠EDC SOHCAHTOA indicates that , so sin This triangle is one of the Pythagorean triples, so CD = 5, and sinθ = = 0.8, which is (D) 10 A If you know your Pythagorean triples, you know that b = 5 You can always use the Pythagorean Theorem to find it as well If x is the smallest angle, it will be opposite the smallest side; in this case, opposite the side with length Secant is the inverse operation of cosine, so start by using SOHCAHTOA to find that For secant, you want 13 E To find tan2 x you can use the identity , which is (A) In this case, Start by using FOIL on the left side of the equation: (1 − sinx)(1 + sinx) = − sin2 x Manipulate the full equation to find that sin2 x = 0.835 Use the trigonometric identity sin2 θ + cos2 θ = 1 to substitute sin2 x = 0.835, and you get cos2 x = 0.165 To find tan2 x, divide sin2 x by , which is (E) 20 C For Statement I, remember that sec and csc , so you really need to know whether cosx = sinz Because the side adjacent x is the same side which is opposite z, this is true; eliminate (B) and (D) For Statement II, you not know the relative values of x and z, so you have no idea whether they are equal; eliminate (E) Finally, to compare sinx to tan x, use SOHCAHTOA: sinx is and tanx is O is the same in both fractions, and in any given right triangle the hypotenuse will be larger than either leg Therefore, because its denominator is greater and the numerators are equal, sinx will be less than tanx; III is true, so choose (C) 25 C This is a right triangle question Create a triangle using Carl’s eye level, the tip of the rocket, and a point on the rocket 1.6 m off the ground The hypotenuse travels through Carl’s eye, and to keep Carl’s body 500 m from the rocket, you need to make sure his eye is also that far from the rocket Because Carl’s eye is 1.6 m above ground level and the rocket is 150 m above ground level, that leg of the triangle should be 150 − 1.6 = 148.4 m, NOT 150 m: For this triangle, you know the sides opposite and adjacent to the x To find the value of x, use tan using 34 C , then solve for x , which is (C) Because you know three sides, you can use the Law of Cosines: In this case, because you want to find ∠A, make a your “c” value and ∠A your C (Remember, in the Law of Cosines c is the side opposite ∠C.) Therefore, to find cosA, the equation becomes: , and solving for cosA you find 0.873 = cosA To find secant, you need ; therefore , which is (C) 35 B Plug In The Answers! Be sure your calculator is in radians, then start trying the answers! If x = , then , and Eliminate (C) It can be difficult to determine whether you need a greater or lesser value of x, so just pick a direction If you try (B), then , and cos = 0.866; choose (B) 40 E Start by drawing the point, including the angle it makes with the x-axis Remember that polar coordinates are in the form (r, θ), where r is the distance from the origin, and θ is the angle in radians Start by using the Pythagorean Theorem to find r: 62 + 82 = r2, r = 10 Eliminate (A) Next, to find θ, you can use Your calculator will tell you that However, this angle is in the fourth quadrant, and point (−8, 6) is in the second quadrant Eliminate (C) You need to find an equivalent angle for which tangent equals − Because tangent is a periodic function which repeats every π units, there is an equivalent angle π units away Therefore, if you add π to −0.644, you get the equivalent angle in the second quadrant, which is 2.498 Eliminate (B) If r = −10, then the angle is measured from its equivalent in the fourth quadrant, which as we saw earlier is equal to −0.644 Neither (D) nor (E) has a θ = −0.644 However, you can find the angle with the same terminal side by adding 2π When you add 2π to −0.644, you get 5.639, so choose (E) 45 D First, the value of cosine must always be between −1 and inclusive, so eliminate (E) Next, if −90° ≤ θ ≤ 90°, then θ is in either the first or fourth quadrants Cosine is positive in both of those quadrants; eliminate (A) and (B) Cosecant is , so Solving for sinθ, you find that sinθ = −0.643 If you take the inverse sine of −0.643 you get −40⁰, and cos −40° = 0.766, which is (D) 47 C Use the trigonometric identity sin2 θ + cos2 θ = 1 and substitute − cos2 θ for each sin2 θ in the equation: Next, subtract from both sides of the equation to isolate the cos2 θ terms: Finally, multiply both sides by −1: 48 E , (C) Be sure your calculator is in radians! Begin by taking the inverse sine of 0.782 to find x: sin−1 0.782 = 0.898 This is NOT between and π; you need to find the equivalent angle in the given range The value of sine starts at 0, increases to at , and decreases symmetrically to 0 at π Therefore, sine at 0.898 past will be the same as sine at 0.898 radians before π To find x you must subtract 0.898 from p to find the equivalent angle, which is 2.244 Finally, find cosine of 3 times 2.244: cos (3(2.244)) = 0.901, which is (E) ... putting everything into either sin or cos You’ll notice that almost everything is over cosx, so rewrite the 1s in the denominator as : Now that all the terms in each set of parentheses are over the. .. denominator, you can combine the numerators within each set of parentheses: Multiply the fractions in the numerator and denominator: Divide by multiplying the numerator by the reciprocal of the. .. with the x-axis Remember that polar coordinates are in the form (r, θ), where r is the distance from the origin, and θ is the angle in radians Start by using the Pythagorean Theorem to find r: 62 +