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Trigonometry

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Angles add to 180° The angles of a triangle always add up to 180°

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Right triangles We only care about right triangles A right triangle is one in which one of the angles is 90° Here’s a right triangle:

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Example. Solve the equations: a) cos (x / 5) = 1 Decision: A) This time we proceed directly to the calculation of the roots of the equation at once: x / 5 = ± arccos (1) + 2πk. Then x / 5 = πk => x = 5πk The answer is: x = 5πk, where k is an integer.

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The Pythagorean Theorem If you square the length of the two shorter sides and add them, you get the square of the length of the hypotenuse adj2 + opp2 = hyp2

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5-12-13 There are few triangles with integer sides that satisfy the Pythagorean formula 3-4-5 and its multiples (6-8-10, etc.) are the best known 5-12-13 and its multiples form another set

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Ratios Since a triangle has three sides, there are six ways to divide the lengths of the sides Each of these six ratios has a name (and an abbreviation) Three ratios are most used: sine = sin = opp / hyp cosine = cos = adj / hyp tangent = tan = opp / adj The other three ratios are redundant with these and can be ignored

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Example. Solve the equations: cos (4x) = √2 / 2. And find all the roots on the interval [0; Π]. Decision: Let us solve our equation in general form: 4x = ± arccos (√2 / 2) + 2πk 4x = ± π / 4 + 2πk; X = ± π / 16 + πk / 2; Now let's see what roots get into our segment.

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Using the ratios With these functions, if you know an angle (in addition to the right angle) and the length of a side, you can compute all other angles and lengths of sides

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Solve equations: а) cos(x/5)=1 The answer is: x = 5πk, where k is an integer.

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Decision: A) This time we proceed directly to the calculation of the roots of the equation at once: Decision: A) This time we proceed directly to the calculation of the roots of the equation at once: x / 5 = ± arccos (1) + 2πk. Then x / 5 = πk => x = 5πk The answer is: x = 5πk, where k is an integer.

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The hard part If you understood this lecture, you’re in great shape for doing all kinds of things with basic graphics Here’s the part I’ve always found the hardest: Memorizing the names of the ratios sin = opp / hyp cos = adj / hyp tan = opp / adj