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Trigonometry Lesson 1: Primary Trigonometric Ratios Todays Objectives: Students will be able to develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles, including: o Identify the hypotenuse of a right triangle and the opposite and adjacent sides for a given acute angle in the triangle o Explain the relationships between similar right triangles and the definitions of the primary trigonometric ratios Let’s begin with a short review of some procedures that are common in trigonometry problems before we discuss the primary trigonometric ratios. Right Triangles A _______ triangle has a right (90º) angle. The other two angles are each _______(between 0º and 90º). The side _______ the right angles is the longest side, and is called the ____________. The two sides__________ to the right angle are called _____ of the right triangle. The word adjacent means “_________”. Pythagorean Theorem In any right triangle, the ________ of the hypotenuse is equal to the _____ of the squares of the other two _______ (legs). This can be illustrated as follows: Note: __________ are often labeled with _________ letters, and the sides__________ the vertices are labeled with the corresponding _______ case letters. We normally label the hypotenuse as ____. Example) Solve for the unknown side length, x. Solution: Sum of the Angles in a Triangle In any triangle, the sum of the measures of the three angles is always equal to 180º Example) Determine the measure of angle XYZ. (When the name of the angle is given as three letters, the middle letter represents the vertex of the angle) Sine, Cosine and Tangent Ratios The three primary trigonometric _______ describe the ratios of the different ______ in a right triangle.These ratios use one of the _______ angles as a point of reference. The 90º angle is ______ used. In the following illustration, the ratios are described relative to _________. Notice that the abbreviations for sine, cosine, and tangent are ____, ____, and ____. SOHCAHTOA You can use the acronym SOH-CAH-TOA to remember these ratios Example Determine the three primary trigonometric ratios from angle θ Solution: first, find the unknown side x Now, write the three ratios Trigonometric Ratios and Similar Triangles __________triangles are triangles in which the corresponding _______ have the same _________. The corresponding _____ in similar triangles are ________________. One way of constructing similar right triangles is shown in the given diagram below. Angles with the same markings have the same measure. Three similar triangles have been formed: ∆AB1C1, ∆AB2C2, ∆AB3C3 Using each of these triangles, consider the ____ ratio for __________. Because the sides are proportional, the sin ratios using each of the three similar triangles are _______. Sin A = opposite/hypotenuse = B1C1/AB1 =B2C2/AB2 = B3C3/AB3 Example) Complete the following steps for the three similar triangles formed in the given diagram. 1) State the tangent of angle θ using the labels of the sides 2) Use a metric ruler to measure the side lengths for each triangle, and give an estimate of the value of tan θ to the nearest hundredth 3) Calculate the value of angle θ Note: Calculators need to be set to Degree Mode, not Radian Mode. Inverse functions are available by using the shift or 2nd function key on your calculator.