Right Triangle Trigonometry Obea Rizzi B Omboy Pythagorean

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. • 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 b c 2 = a 2 + b 2

Similar Triangles are similar if two conditions are met: 1. The corresponding angle measures are equal. 2. 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 B a C e d F

Corresponding Angles and 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

Example Find the missing side lengths for the similar triangles. 3. 2 3. 8 y 54. 4 x 42. 5

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.

Introduction to Trigonometry In this section we define 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 hyp opp adj A

Definitions Sine is abbreviated sin, cosine is 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 Opp Hyp Cos Adj Hyp Tan Opp Adj

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 2 √ 3 1 60°

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

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 45° cos(45°) = 1/√ 2 tan(45°) = 1 1 √ 2 1 45°

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