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Right Triangle Trigonometry

Right Triangle Trigonometry. Obea Rizzi B. Omboy. Pythagorean Theorem. Recall that a right triangle has a 90 ° angle as one of its angles. The side that is opposite the 90 ° angle is called the hypotenuse .

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Right Triangle Trigonometry

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  1. Right Triangle Trigonometry ObeaRizzi B. Omboy

  2. Pythagorean Theorem • Recall that a right triangle has a 90° angle as one of its angles. • The side that is opposite the 90° angle is called the hypotenuse. • The theorem due to Pythagoras says that the square of the hypotenuse is equal to the sum of the squares of the legs. a c c2 = a2 + b2 b

  3. Similar Triangles Triangles are similar if two conditions are met: • The corresponding angle measures are equal. • Corresponding sides must be proportional. (That is, their ratios must be equal.) The triangles below are similar. They have the same shape, but their size is different. A D c b f e E d F B a C

  4. Corresponding Anglesand Sides As you can see from the previous page we can see that angle A is equal to angle D, angle B equals angle E, and angle C equals angle F. The lengths of the sides are different but there is a correspondence. Side a is in correspondence with side d. Side b corresponds to side e. Side c corresponds to side f. What we do have is a set of proportions a/d= b/e = c/f

  5. Example Find the missing side lengths for the similar triangles. 3.2 3.8 y 54.4 x 42.5

  6. ANSWER • Notice that the 54.4 length side corresponds to the 3.2 length side. This will form are complete ratio. • To find x, we notice side x corresponds to the side of length 3.8. • Thus we have 3.2/54.4 = 3.8/x. Solve for x. • Thus x = (54.4)(3.8)/3.2 = 64.6 • Same thing for y we see that 3.2/54.4 = y/42.5. Solving for y gives y = (42.5)(3.2)/54.4 = 2.5.

  7. Introduction to Trigonometry In this section we define the three basic trigonometric ratios, sine, cosine and tangent. • opp is the side opposite angle A • adj is the side adjacent to angle A • hyp is the hypotenuse of the right triangle hypopp adjA

  8. Definitions Sineis abbreviated sin, cosineis abbreviated cos and tangent is abbreviated tan. • The sin(A) = opp/hyp • The cos(A) = adj/hyp • The tan(A) = opp/adj • Just remember sohcahtoa! • Sin OppHypCos AdjHypTan OppAdj

  9. Special Triangles Special triangle is a triangle with 30 – 60 – 90 degree measurement in its angles. Consider an equilateral triangle with side lengths 2. Recall the measure of each angle is 60°. Chopping the triangle in half gives the 30 – 60 – 90 degree triangle. 30° 2 2 √3 1 2 60°

  10. 30° – 60° – 90° Now we can define the sine cosine and tangent of 30° and 60°. • sin(60°)=√3 / 2; cos(60°) = ½; tan(60°) = √3 • sin(30°) = ½ ; cos(30°) = √3 / 2; tan(30°) = 1/√3

  11. 45° – 45° – 90° Consider a right triangle in which the lengths of each leg are 1. This implies the hypotenuse is √2. sin(45°) = 1/√2 cos(45°) = 1/√2 tan(45°) = 1 45° 1 √2 1 45°

  12. Example Find the missing side lengths and angles. A = 180°-90°-60°=30° sin(60°)=y/10 thus y = 10sin(60°) 60° 10 x A y

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