Honors Geometry Sections 10.1 & 10.2 Trigonometric ratios The word “trigonometry” comes from two Greek words which mean ___________________ triangle measurement And that is exactly what we will do with trigonometry - we will find the measures of angles and the length of sides in triangles. Initially, we will consider only right triangles, but later in this section, we will consider trigonometry involving acute and obtuse triangles. A trigonometric ratio (i.e. fraction) is a ratio of the lengths of two sides of a right triangle. The three basic trigonometric ratios are sine (___), sin cosine (___) cos and tangent (___). tan These ratios are defined for the acute angles of a right triangle as follows. opposite leg sin A = -------------hypotenuse adj. hyp. opp. adjacent leg cos A = -------------hypotenuse opposite leg tan A = adjacent ------------leg If we write the ratios for the acute angle B instead, then the opposite leg and adjacent leg would be switched!!!! An easy way to remember these three trig ratios is with the mnemonic SOH-CAH-TOA A P D I P Y O D Y N P E P S J P N P J Example: Give the three trig ratios for the following triangles. hyp opp adj 3 5 4 5 3 4 Example: Give the three trig ratios for the following triangles. 5 5 10 10 5 2 5 Sin O = 25 5 5 5 10 10 5 2 5 Cos G = 25 5 5 5 5 1 Tan G = 10 2 5 10 c 2 2 2 c 125 2 c 125 5 5 The value of the sine, cosine and tangent of an angle depend only upon the measure of the angle and not the size of the triangle that the angle is found in. Here’s why!!! ACE ~ BCD by _____ AA Therefore, AC AE CE BC BD CD ACE ~ BCD by _____ AA Therefore, AC BC AE CE BD CD Now, we can take the first two ratios separately and AE rearrange the terms to get BD . Note: These are the BC AC ratios for sin C in both ACE and BCD . Similarly, we can take the first and third ratios CD CE separately and rearrange the terms to get BC AC . Note: These are the ratios for cos C in both ACE and BCD . A scientific calculator can be used to find the value of these three trig ratios. Make sure your calculator is in degree mode. .225 sin 130 = ___________ .225 tan 400 = ____________ .839 cos 770 = ____________ We can use the trig ratios to find the lengths of unknown sides in right triangles. Example: Solve for x and y. Round your answers to the nearest 1000th. 12 sin 28 x x sin 28 12 12 x sin 28 x 25.561 tan 28 12 y y tan 28 12 12 y tan 28 y 22.569 Example: Solve for x and y. Round your answers to the nearest 1000th. x cos 40 20 20 cos 40 x x 15.321 y sin 40 20 20 sin 40 y y 12.856 Example: The angle of elevation to the top of a tree from a point 100 feet from the base of the tree is 510. Estimate the height of the tree to the nearest 1000th. NOTE: the angle of elevation is the angle formed by a horizontal line (usually the ground) and the line of sight up to some object. Example: The angle of elevation to the top of a tree from a point 100 feet from the base of the tree is 510. Estimate the height of the tree to the nearest 1000th. x tan 51 100 100 tan 51 x x 123.490 ft x 51 100