Primary Trigonometric Ratios - Jon Garvin

trigonometryMPM2D: Principles of MathematicsPrimary Trigonometric RatiosJ. GarvinSlide 1/10

trigonometrySimilar TrianglesIn the diagram below, ABC ADE since A is commonto both triangles, and ACB AED.J. Garvin — Primary Trigonometric RatiosSlide 2/10

trigonometrySimilar TrianglesIn the diagram below, ABC ADE since A is commonto both triangles, and ACB AED.This means that any ratio of two sides in ABC is equal tothe ratio of corresponding sides in ADE .J. Garvin — Primary Trigonometric RatiosSlide 2/10

trigonometrySimilar TrianglesBy varying the measure of A, the ratio of two sides in ABC will change, but will remain equal to the ratio ofcorresponding sides in ADE .J. Garvin — Primary Trigonometric RatiosSlide 3/10

trigonometrySimilar TrianglesBy varying the measure of A, the ratio of two sides in ABC will change, but will remain equal to the ratio ofcorresponding sides in ADE .Therefore, a specific measure of A can be associated with aspecific ratio of two sides in a right triangle.J. Garvin — Primary Trigonometric RatiosSlide 3/10

trigonometrySimilar TrianglesBy varying the measure of A, the ratio of two sides in ABC will change, but will remain equal to the ratio ofcorresponding sides in ADE .Therefore, a specific measure of A can be associated with aspecific ratio of two sides in a right triangle.Is the ratio of two sides associated with a given angle unique?J. Garvin — Primary Trigonometric RatiosSlide 3/10

trigonometrySimilar TrianglesBy varying the measure of A, the ratio of two sides in ABC will change, but will remain equal to the ratio ofcorresponding sides in ADE .Therefore, a specific measure of A can be associated with aspecific 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.J. Garvin — Primary Trigonometric RatiosSlide 3/10

trigonometrySimilar TrianglesBy varying the measure of A, the ratio of two sides in ABC will change, but will remain equal to the ratio ofcorresponding sides in ADE .Therefore, a specific measure of A can be associated with aspecific 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 alsoincreases. Thus, the ratio will increase.J. Garvin — Primary Trigonometric RatiosSlide 3/10

trigonometrySimilar TrianglesBy varying the measure of A, the ratio of two sides in ABC will change, but will remain equal to the ratio ofcorresponding sides in ADE .Therefore, a specific measure of A can be associated with aspecific 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 alsoincreases. Thus, the ratio will increase.If A decreases, the length of the opposite side alsodecreases. Thus, the ratio will decrease.J. Garvin — Primary Trigonometric RatiosSlide 3/10

trigonometrySimilar TrianglesBy varying the measure of A, the ratio of two sides in ABC will change, but will remain equal to the ratio ofcorresponding sides in ADE .Therefore, a specific measure of A can be associated with aspecific 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 alsoincreases. Thus, the ratio will increase.If A decreases, the length of the opposite side alsodecreases. 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 RatiosSlide 3/10

trigonometryPrimary Trigonometric RatiosIn the right triangle ABC below, the three sides have beenlabelled based on their position relative to A.J. Garvin — Primary Trigonometric RatiosSlide 4/10

trigonometryPrimary Trigonometric RatiosIn the right triangle ABC below, the three sides have beenlabelled based on their position relative to A.The opposite and adjacent sides are reversed relative to B,but the hypotenuse is always across from the right angle.J. Garvin — Primary Trigonometric RatiosSlide 4/10

trigonometryPrimary Trigonometric RatiosThere are six possible ratios of sides that can be made fromthe three sides.J. Garvin — Primary Trigonometric RatiosSlide 5/10

trigonometryPrimary Trigonometric RatiosThere are six possible ratios of sides that can be made fromthe three sides.The three primary trigonometric ratios are sine, cosine andtangent.J. Garvin — Primary Trigonometric RatiosSlide 5/10

trigonometryPrimary Trigonometric RatiosThere are six possible ratios of sides that can be made fromthe three sides.The three primary trigonometric ratios are sine, cosine andtangent.Primary Trigonometric RatiosLet ABC be a right triangle with A 6 90 . Then, thethree primary trigonometric ratios for A are:oppositeSine: sin A hypotenuseadjacentCosine: cos A hypotenuseoppositeTangent: tan A adjacentJ. Garvin — Primary Trigonometric RatiosSlide 5/10

trigonometryPrimary Trigonometric RatiosThere are six possible ratios of sides that can be made fromthe three sides.The three primary trigonometric ratios are sine, cosine andtangent.Primary Trigonometric RatiosLet ABC be a right triangle with A 6 90 . Then, thethree primary trigonometric ratios for A are:oppositeSine: sin A hypotenuseadjacentCosine: cos A hypotenuseoppositeTangent: tan A adjacentThe phrase SOH-CAH-TOA is a mnemonic for these ratios.J. Garvin — Primary Trigonometric RatiosSlide 5/10

trigonometryPrimary Trigonometric RatiosExampleState the three primary trigonometric ratios for A in ABC .J. Garvin — Primary Trigonometric RatiosSlide 6/10

trigonometryPrimary Trigonometric RatiosExampleState the three primary trigonometric ratios for A in ABC .sin A J. Garvin — Primary Trigonometric RatiosSlide 6/10opphyp35

trigonometryPrimary Trigonometric RatiosExampleState the three primary trigonometric ratios for A in ABC .sin A cos A J. Garvin — Primary Trigonometric RatiosSlide 6/10opphyp35adjhyp45

trigonometryPrimary Trigonometric RatiosExampleState the three primary trigonometric ratios for A in ABC .sin A cos A tan A J. Garvin — Primary Trigonometric RatiosSlide 6/10opphyp35adjhyp45oppadj34

trigonometryPrimary Trigonometric RatiosExampleState the three primary trigonometric ratios for A in ABC . Express all ratios in simplest form.J. Garvin — Primary Trigonometric RatiosSlide 7/10

trigonometryPrimary Trigonometric RatiosExampleState the three primary trigonometric ratios for A in ABC . Express all ratios in simplest form.sin A J. Garvin — Primary Trigonometric RatiosSlide 7/10opphyp513

trigonometryPrimary Trigonometric RatiosExampleState the three primary trigonometric ratios for A in ABC . Express all ratios in simplest form.sin A cos A J. Garvin — Primary Trigonometric RatiosSlide 7/10opphyp513adjhyp1213

trigonometryPrimary Trigonometric RatiosExampleState the three primary trigonometric ratios for A in ABC . Express all ratios in simplest form.sin A cos A tan A J. Garvin — Primary Trigonometric RatiosSlide 7/10opphyp513adjhyp1213oppadj512

trigonometryPrimary Trigonometric RatiosExampleState the three primary trigonometric ratios for A in ABC .J. Garvin — Primary Trigonometric RatiosSlide 8/10

trigonometryPrimary Trigonometric RatiosExampleState the three primary trigonometric ratios for A in ABC .Use the PythagoreanTheorem to determinethe length of thehypotenuse, h.h2 62 32h2 45 h 45J. Garvin — Primary Trigonometric RatiosSlide 8/10

trigonometryPrimary Trigonometric RatiosThis gives us the following right triangle.J. Garvin — Primary Trigonometric RatiosSlide 9/10

trigonometryPrimary Trigonometric RatiosThis gives us the following right triangle.sin A opphyp 345J. Garvin — Primary Trigonometric RatiosSlide 9/10

trigonometryPrimary Trigonometric RatiosThis gives us the following right triangle.sin A opphypcos A adjhyp 345 645J. Garvin — Primary Trigonometric RatiosSlide 9/10

trigonometryPrimary Trigonometric RatiosThis gives us the following right triangle.oppsin A hyp 345J. Garvin — Primary Trigonometric RatiosSlide 9/10adjcos A hyp 645tan A oppadj3612

trigonometryQuestions?J. Garvin — Primary Trigonometric RatiosSlide 10/10

Primary Trigonometric Ratios There are six possible ratios of sides that can be made from the three sides. The three primary trigonometric ratios are sine, cosine and tangent. Primary Trigonometric Ratios Let ABC be a right triangle with \A 6 90 . Then, the three primary trigonometric ratios for \A are: Sine: sinA opposite hypotenuse Cosine .