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trigonometry trigonometry Similar Triangles MPM2D: Principles of Mathematics In the diagram below, ∆ABC ∼ ∆ADE since ∠A is common to both triangles, and ∠ACB = ∠AED. Primary Trigonometric Ratios J. Garvin This means that any ratio of two sides in ∆ABC is equal to the ratio of corresponding sides in ∆ADE . J. Garvin — Primary Trigonometric Ratios Slide 2/10 Slide 1/10 trigonometry trigonometry Similar Triangles Primary Trigonometric Ratios By varying the measure of ∠A, the ratio of two sides in ∆ABC will change, but will remain equal to the ratio of corresponding sides in ∆ADE . In the right triangle ∆ABC below, the three sides have been labelled based on their position relative to ∠A. Therefore, a specific measure of ∠A can be associated with a specific ratio of two sides in a right triangle. Is the ratio of two sides associated with a given angle unique? Consider the ratio of the opposite side to the hypotenuse. If ∠A increases, the length of the opposite side also increases. Thus, the ratio will increase. If ∠A decreases, the length of the opposite side also decreases. Thus, the ratio will decrease. In both scenarios, the ratio changes with the measure of ∠A. Therefore, the ratio associated with a specific angle is unique. J. Garvin — Primary Trigonometric Ratios Slide 3/10 The opposite and adjacent sides are reversed relative to ∠B, but the hypotenuse is always across from the right angle. J. Garvin — Primary Trigonometric Ratios Slide 4/10 trigonometry trigonometry Primary Trigonometric Ratios Primary Trigonometric Ratios There are six possible ratios of sides that can be made from the three sides. Example The three primary trigonometric ratios are sine, cosine and tangent. State the three primary trigonometric ratios for ∠A in ∆ABC . sin A = Primary Trigonometric Ratios Let ∆ABC be a right triangle with ∠A 6= 90◦ . Then, the three primary trigonometric ratios for ∠A are: opposite Sine: sin A = hypotenuse adjacent Cosine: cos A = hypotenuse opposite Tangent: tan A = adjacent = cos A = = tan A = = The phrase SOH-CAH-TOA is a mnemonic for these ratios. J. Garvin — Primary Trigonometric Ratios Slide 5/10 J. Garvin — Primary Trigonometric Ratios Slide 6/10 opp hyp 3 5 adj hyp 4 5 opp adj 3 4 trigonometry trigonometry Primary Trigonometric Ratios Primary Trigonometric Ratios Example Example State the three primary trigonometric ratios for ∠A in ∆ABC . Express all ratios in simplest form. State the three primary trigonometric ratios for ∠A in ∆ABC . Use the Pythagorean Theorem to determine the length of the hypotenuse, h. sin A = = cos A = = tan A = = opp hyp 5 13 adj hyp h2 = 62 + 32 h2 = 45 √ h = 45 12 13 opp adj 5 12 J. Garvin — Primary Trigonometric Ratios Slide 8/10 J. Garvin — Primary Trigonometric Ratios Slide 7/10 trigonometry Primary Trigonometric Ratios trigonometry Questions? This gives us the following right triangle. sin A = opp hyp cos A = adj hyp = √3 45 = √6 45 J. Garvin — Primary Trigonometric Ratios Slide 9/10 tan A = = = opp adj 3 6 1 2 J. Garvin — Primary Trigonometric Ratios Slide 10/10