Lesson 1: Primary Trigonometric Ratios 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: Identify the hypotenuse of a right triangle and the opposite and adjacent sides for a given acute angle in the triangle Explain the relationships between similar right triangles and the definitions of the primary trigonometric ratios

Right Triangles A right triangle has a right (90º) angle. The other two angles are acute (between 0º and 90º). The side opposite the right angle is the longest side, and is called the hypotenuse. The two sides adjacent to the right angle are called legs of the right triangle. The word adjacent means “beside”. Hypotenuse Legs Acute Angles Right Angle

Pythagorean Theorem In any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs). This can be illustrated as follows: 𝑐 2 = 𝑎 2 + 𝑏 2 Note: Vertices are often labeled with capital letters, and the sides opposite the vertices are labeled with the corresponding lower case letters. We normally label the hypotenuse as c. A c b B C a

Example Solve for the unknown side length, x. (a = 5, b = x, c = 11) Solution: Apply the pythagorean theorem, 𝑐 2 = 𝑎 2 + 𝑏 2 𝑏 2 = 𝑐 2 − 𝑎 2 , 𝑏= 𝑐 2 − 𝑎 2 𝑏= 11 2 − 5 2 = 121−25 = 96 ≈9.8 𝑐𝑚 11 cm 5 cm x

Sum of the Angles in a Triangle In any triangle, the sum of the measures of the three angles is always equal to 180º B 100° 45° 35° A C

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) Solution: 180°- 85° - 47° = 48° X 85° 47° Y Z

Sine, Cosine, and Tangent Ratios The three primary trigonometric ratios describe the ratios of the different sides in a right triangle. These ratios use one of the acute angles as a point of reference. The 90º angle is never used. In the following illustration, the ratios are described relative to angle θ. Notice that the abbreviations for sine, cosine, and tangent are sin, cos, and tan. hypotenuse Opposite side to θ θ Adjacent side to θ

SOHCAHTOA SOHCAHTOA You can use the acronym SOH-CAH-TOA to remember these ratios 𝑠𝑖𝑛𝜃= 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑐𝑜𝑠𝜃= 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑡𝑎𝑛𝜃= 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

Example Determine the three primary trigonometric ratios from angle θ Solution: first, find the unknown side x 𝑥= 13 2 − 12 2 = 25 = 5 𝑐𝑚 Now, write the three ratios 𝑠𝑖𝑛𝜃= 12 13 𝑐𝑜𝑠𝜃= 5 13 𝑡𝑎𝑛𝜃= 12 5 θ 13 cm x 12 cm

Trigonometric Ratios and Similar Triangles Similar triangles are triangles in which the corresponding angles have the same measure. The corresponding sides in similar triangles are proportional. One way of constructing similar right triangles is shown in the given diagram below. B3 26 cm B2 29 cm 35 cm B1 37 cm 25 cm 14 cm A C3 C2 C1

Similar Triangles 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 sine ratio for angle A. Because the sides are proportional, the sine ratios using each of the three similar triangles are equal. sin 𝐴= 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝐵1𝐶1 𝐴𝐵1 = 14 37 ≈0.378 𝐵2𝐶2 𝐴𝐵2 = 25 66 ≈0.379 𝐵3𝐶3 𝐴𝐵3 = 35 92 ≈0.380

Example 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. B3 B2 B1 C3 C2 C1 A

Solution 1) 𝑡𝑎𝑛𝜃= 𝐴𝐶1 𝐶1𝐵1 𝑡𝑎𝑛𝜃= 𝐴𝐶2 𝐶2𝐵2 𝑡𝑎𝑛𝜃= 𝐴𝐶3 𝐶3𝐵3 2) 𝑡𝑎𝑛𝜃≅ 1) 𝑡𝑎𝑛𝜃= 𝐴𝐶1 𝐶1𝐵1 𝑡𝑎𝑛𝜃= 𝐴𝐶2 𝐶2𝐵2 𝑡𝑎𝑛𝜃= 𝐴𝐶3 𝐶3𝐵3 2) 𝑡𝑎𝑛𝜃≅ 3) 𝜃≅