Cracking the SAT Subject Test in Math 2, 2nd Edition 5 TR = ? ∠S = ? ∠T = ? 6 YW = ? ∠W = ? ∠Y = ? TRIGONOMETRIC IDENTITIES On the SAT Subject Test in Math 2, sometimes you will be asked to work an al[.]
5 TR = ? ∠S = ? ∠T = ? YW = ? ∠W = ? ∠Y = ? TRIGONOMETRIC IDENTITIES On the SAT Subject Test in Math 2, sometimes you will be asked to work an algebraic problem using trigonometry These questions often boil down to using SOHCAHTOA in a handful of clever ways Often, the way to simplify equations that are mostly made up of trigonometric functions is to express the functions as follows: Writing trig functions this way can simplify trig equations, as the following example shows: Working with trig functions this way lets you simplify expressions The equation above is actually a commonly used trigonometric identity You should memorize this, as it can often be used to simplify equations Here’s the breakdown of another frequently used trigonometric identity: That last step may seem a little baffling, but it’s really simple This equation is based on a right triangle, in which O and A are legs of the triangle, and H is the hypotenuse Consequently you know that O2 + A2 = H2 That’s just the Pythagorean Theorem That’s what lets you do the last step, in which This completes the second commonly used identity that you should memorize sin2θ +cos2θ = 1 In addition to memorizing these two identities, you should practice working algebraically with trig functions in general Some questions may require you to use the SOHCAHTOA definitions of the trig functions; others may require you to use the two identities you’ve just reviewed Take a look at these examples: 20 If sinx = 0.707, then what is the value of (sinx) × (cosx) × (tanx) ? (A) 1.0 (B) 0.707 (C) 0.5 (D) 0.4 (E) 0.207 Here’s How to Crack It This is a tricky question To solve it, simplify that complicated trigonometric expression Writing in the SOHCAHTOA definitions works just fine, but in this case it’s even faster to use one of those identities Now it’s a simpler matter to answer the question If sinx = 0.707, then sin2x = 0.5 The answer is (C) Take a look at this one: 21 If sina = 0.4, and 1 − cos2a = x, then what is the value of x ? (A) 0.8 (B) 0.6 (C) 0.44 (D) 0.24 (E) 0.16 Here’s How to Crack It Here again, the trick to the question is simplifying the complicated trig expression Since sin2θ + cos2θ = 1, you can rearrange any of those terms to rephrase it Using the second trig identity, you can quickly take these steps: 1 − cos2a = x sin2a = x (0.4)2 =x x = 0.16 And that’s the answer Choice (E) is correct Let’s look at another problem: 38 If = 0.345, then sin2x = (A) −0.655 (B) −0.345 (C) 0.345 (D) 0.655 (E) Here’s How to Crack It You know that tan So you can rework the problem: If you multiply by the reciprocal of the denominator, you get Now, to get to sin2x, you need to replace cos2x with 1 − sin2x: 1 − sin2x = 0.345 −sin2x = −0.655 sin2x = 0.655 The answer is (D) Plugging In on Trigonometric Algebra As you saw in the Algebra chapter, Plugging In is a powerful tool on many SAT Subject Test in Math problems This is also true for problems dealing with trigonometric identities: just Plug In for the unknown angle When you Plug In on these problems, choose a value for x or θ which gives unique values for sin, cos, and tan Avoid multiples of 45 (such as 45, 90, or 180) if you’re in degrees or multiples of in radians Try one more problem: 26 = (A) secx (B) sin2x (C) cosx (D) tanx (E) cotx Here’s How to Crack It Make sure your calculator is in degrees mode and make x = 20 Use your calculator to find the value of the expression (remember that to find cos2 20° you need to find cos20° and then square the result): Now, you can Plug In x = 20° to each answer choice and find the choice which equals 2.747 You don’t need to check (B) and (C), as they cannot be greater than 1 Also remember that secx = and cotx = The only choice which is close to 2.744 is (E) (Note: you may have noticed that 1 − cos2x = sinxx, which lets you cancel sinx from the numerator and denominator, leaving you with , which is cot x However, when you DON’T notice these things, or have a moment of bafflement on the test, Plugging In can get you to the answer.) Now, use what you’ve learned about SOHCAHTOA and trigonometric identities to simplify the trigonometric expressions in the following problems DRILL 3: TRIGONOMETRIC IDENTITIES Try the following practice questions The answers can be found in Part IV 10 (1 − sinx)(1 + sinx) = (A) cosx (B) sinx (C) tanx (D) cos2x (E) sin2x 16 = (A) (B) (C) (D) cos2x (E) tanx 24 − (sinx)(tanx) = (A) cosx (B) sinx (C) tanx (D) cos2x (E) sin2x 38 = (A) 1 − cosx (B) 1 − sinx (C) tanx + 1 (D) cos2x (E) sin2x 45 = (A) (B) 0.5 (C) sin2θ (D) cos2θ (E) sinθ cosθ The Other Trig Functions On the SAT Subject Test in Math 2, you may run into the other three trigonometric functions—the cosecant, secant, and cotangent These functions are abbreviated cscθ, secθ, and cotθ, respectively, and they are simply the reciprocals of the three basic trigonometric functions you’ve already reviewed Here’s how they relate: You can also express these functions in terms of the sides of a right triangle—just by flipping over the SOHCAHTOA definitions of the three basic functions These three functions generally show up in algebra-style questions, which require you to simplify complex expressions containing trig functions The goal is usually to get an expression into the simplest form possible, one that contains no fractions Such questions are like algebra-style questions involving the three basic trig functions; the only difference is that the addition of three more functions increases the number of possible forms an expression can take For example: The entire expression (cosx)(cotx) + (sin2x cscx) is therefore equivalent to a single trig function, the cosecant of x That’s generally the way algebraic trigonometry questions work on the SAT Subject Test in Math 2 DRILL 4: OTHER TRIG FUNCTIONS Simplify each of these expressions to a single trigonometric function Keep an eye out for the trigonometric identities reviewed on this page they’ll still come in handy The answers can be found in Part IV 19 sec2x − 1 = (A) sinx cosx (B) sec2x (C) cos2x (D) sin2x (E) tan2x 23 = (A) cosx ... cos2x (E) sin2x 45 = (A) (B) 0.5 (C) sin2θ (D) cos2θ (E) sinθ cosθ The Other Trig Functions On the SAT Subject Test in Math 2, you may run into the other three trigonometric functions? ?the cosecant,... The answer is (D) Plugging In on Trigonometric Algebra As you saw in the Algebra chapter, Plugging In is a powerful tool on many SAT Subject Test in Math problems This is also true for problems dealing with trigonometric identities: just Plug In for the unknown angle... The entire expression (cosx)(cotx) + (sin2x cscx) is therefore equivalent to a single trig function, the cosecant of x That’s generally the way algebraic trigonometry questions work on the SAT Subject Test in Math 2 DRILL 4: OTHER TRIG FUNCTIONS