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Section 11 – 1 Simplifying Radicals • Multiplication Property of Square Roots: For every number a > 0 and b > 0, ab a b • You can multiply numbers that are both under the radical and you can separate a number under a radical into two radicals being multiplied by each other • First 10 perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (know these) • Simplifying a radical is when you rewrite a radical so that all of the perfect squares have been factored out • The end result is said to be in simplified radical form • Simplify each expression • Ex1. 243 • Ex2. 28x7 • Ex3. 12 32 • Ex4. 7 5x 3 8 x • Division Property of Square Roots: For every number a > 0 and b > 0, a a b b • You use this property just as you would with multiplication • Fractions are not allowed to have radicals in the denominator • You must rationalize the denominator • 1) Simplify the radical(s) • 2) Multiply the numerator and denominator by the radical remaining in the denominator • Simplify • Ex5. 75 x5 48 x 3 • Ex6. 7 • Ex7. 11 12x 3 Section 11-2 The Pythagorean Theorem • The Pythagorean Theorem is applied only to RIGHT triangles • You can use this theorem to find the length of missing sides • The two shortest sides of a right triangle are called the legs (they must meet at a 90° angle) • The longest side is called the hypotenuse (it is directly across from the 90° angle) • The Pythagorean Theorem: a² + b² = c² where a and b are the legs and c is the hypotenuse • Find the length of the missing side (to the nearest tenth) • Ex1. a = 8, b = 12, c = ? • Ex2. a = 20, b = ?, c = 41 • Ex3. 67 ft 25 ft x Section 11 – 3 The Distance and Midpoint Formulas • The Pythagorean Distance Formula: The distance d between any two points (x1, y1) and (x2, y2) is d x x y y • Ex1. Find the distance between A(-3, 7) and B(5, -4). Show work. • Ex2. Find the perimeter of ∆XYZ with X(-3, 4) Y(-1, -5) and Z(2, 2). Show work. • The Pythagorean Distance formula can be derived from the Pythagorean Theorem 2 2 1 2 2 1 • If you are asked to give an answer in exact form, you are to give it in simplified radical form • The midpoint of a segment is the point that divides the segment into two equal segments • The Midpoint Formula: The midpoint M of a line segment with endpoints A(x1, y1) and x1 x2 y1 y2 B(x2, y2) is , 2 2 • Ex3. If segment CD has endpoints C(-4, 7) and D(3, -2), find the midpoint of CD. Show work. Section 11 – 4 Operations with Radical Expressions • Radicals are like radicals if they have the same radicand (the same number under the radical symbol) • Unlike radicals have different numbers under the radical • You can add and subtract like radicals, just as you could with like terms • Ex1. Simplify A) 4 5 7 5 B) 3 5 8 20 • You can distribute with radicals as well (remembering that you can multiply radicals together and then simplify them if possible) • Ex2. Simplify 10 6 5 • Ex3. Use FOIL & then simplify 3 2 7 3 4 7 • Conjugates are the sum and the difference of the same two terms (i.e. 3 7 and 3 7 are conjugates) • The product of two conjugates results in the difference of two squares (an integer) • To rationalize the denominator of an expression that has an addition or subtraction radical expression in the denominator, you must multiply the numerator and denominator by the conjugate of the denominator 5 • Ex4. Rationalize the denominator 3 11 • You should never leave a radical in the denominator of the a fraction! Section 11 – 5 Solving Radical Equations • A radical equation is an equation that has a variable as a radicand • Remember that the expression under a radical must be nonnegative • Ex1. Solve each equation. a) a 8 6 b) x 7 9 • If an equation has radical expressions on both sides, square each side and then solve • Ex2. Solve 3m 7 5m 13 • When you solve an equation by squaring each side, you create a new equation. This new equation may have solutions that do not solve the original equation. See page 609 • These solutions that do not solve the original equation are called extraneous solutions • Ex3. Solve a) m m 8 b) 6 x 9 4 • You should make a table of values to create an accurate graph Section 11 – 6 Graphing Square Root Functions • A square root function is a function that contains the independent variable in the radicand • The parent function for square root functions is y x • The graph of the parent function is the positive half (because radicands can’t be negative) of a sideways parabola (see page 614) • The domain of a function contains all possible values of the independent variable • The domain of the parent function is {x: x > 0} • You can find the domain by graphing and looking at the graph or you can determine algebraically what values can meaningfully be substituted for x • Ex1. Find the domain of y x 2 • The equation y x k is a translation of the parent function by k units up • The equation y x k is a translation of the parent function by k units down • The equation y x h is a translation of the parent function by h units to the left • The equation y x h is a translation of the parent function by h units to the right • Ex2. Graph each equation a) y x 3 b) y x 5 Section 11 – 7 Trigonometric Ratios • There are three trigonometric ratios: sine (sin), cosine (cos), and tangent (tan) • These ratios describe a specific relationship between an angle in a RIGHT triangle and two of the sides of that triangle sin opposite hypotenuse cos adjacent hypotenuse tan opposite adjacent • SOHCAHTOA should help you remember these ratios (if you spell it correctly) • Use a capital letter to represent an angle • Open to page 621 to see how to identify adjacent leg vs. opposite leg vs. hypotenuse • Ex1. Use the triangle below to find a) sin X b) cos X c) tan X X 5 ft 3 ft Y 4 ft Z • You can use your calculator to find the value of trigonometric functions • Make sure your calculator mode is in degrees! • Ex2. Find the value of each expression. Round to the nearest thousandth. a) sin 130° b) cos 130° c) tan 130° • You can use SOHCAHTOA to find the lengths of missing sides of a right triangle • Ex3. Find the length of x. 37° x 29 • An angle of elevation is an angle from the horizontal up to a line of sight (see page 623) • An angle of depression is an angle measured below the horizontal line of sight (see page 624) • You can use angle of elevation and angle of depression with trigonometric functions to solve for missing lengths (see example 4 on page 623 and example 5 on page 624)