Math12 lesson 2


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  • Week 2
  • Math12 lesson 2

    1. 1. Lesson 2: TRIGONOMETRY OF RIGHT TRIANGLES<br />Math 12 <br />Plane and Spherical Trigonometry<br />
    2. 2. OBJECTIVES<br />At the end of the lesson the students are expected to:<br />Define the six trigonometric functions as ratios of the sides of a right triangle<br />Evaluate the trigonometric functions of an angle<br />Evaluate the trigonometric functions of special angles<br />Solve right triangles.<br />
    3. 3. TRIGONOMETRIC FUNCTIONS<br />Let πœƒ be an acute angle in a right triangle, then <br />π‘ π‘–π‘›π‘’π‘ π‘–π‘›πœƒ=π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’Β π‘ π‘–π‘‘π‘’hπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’<br />π‘π‘œπ‘ π‘–π‘›π‘’π‘π‘œπ‘ πœƒ=π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘Β π‘ π‘–π‘‘π‘’hπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’<br />π‘‘π‘Žπ‘›π‘”π‘’π‘›π‘‘π‘‘π‘Žπ‘›πœƒ=π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’Β π‘ π‘–π‘‘π‘’π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘Β π‘ π‘–π‘‘π‘’<br />π‘π‘œπ‘‘π‘Žπ‘›π‘”π‘’π‘›π‘‘π‘π‘œπ‘‘πœƒ=π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘Β π‘ π‘–π‘‘π‘’π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’Β π‘ π‘–π‘‘π‘’<br />π‘ π‘’π‘π‘Žπ‘›π‘‘π‘ π‘’π‘πœƒ=hπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘Β π‘ π‘–π‘‘π‘’<br />π‘π‘œπ‘ π‘’π‘π‘Žπ‘›π‘‘π‘π‘ π‘πœƒ=hπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’Β π‘ π‘–π‘‘π‘’<br />Β <br />hypotenuse<br />Opposite<br /> side<br />Adjacent <br /> side<br />
    4. 4. RECIPROCAL FUNCTIONS<br />The following gives the reciprocal relations of the six trigonometric functions:<br />sinπœƒ=1cscπœƒΒ Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β cscπœƒ=1sinπœƒ<br />cosπœƒ=1secπœƒΒ Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β secπœƒ=1cosπœƒ<br />tanπœƒ=1cotπœƒΒ Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β cotπœƒ=1tanπœƒ<br />Β <br />
    5. 5. PYTHAGOREAN THEOREM<br />The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Referring to the right triangle below, then<br />𝑐2=π‘Ž2+𝑏2<br />The Pythagorean Theorem<br />is used to find the side of<br />a right triangle<br />Β <br />B<br />c<br />a<br />C<br />A<br />b<br />
    6. 6. FUNCTIONS OF COMPLEMENTARY ANGLES<br />B<br />c<br />a<br />sin A = <br />cos B = <br />cos A = <br />sin B = <br />tan A = <br />cot B = <br />tan B = <br />cot A = <br />sec A = <br />csc A = <br />sec B = <br />csc B = <br />C<br />A<br />b<br />Comparing the trigonometric functions of the acute angles A and B, and making use of the fact that A and B are complementary angles (A+B=900), then<br />
    7. 7. FUNCTIONS OF COMPLEMENTARY ANGLES<br />sin B = sin = cos A <br />cos B = cos = sin A <br />tan B = tan = cot A <br />cot B = cot = tan A <br />csc B = csc = sec A <br />sec B = sec = csc A <br />The relations may then be expressed by a single statement that: A trigonometric function of an angle is always equal to the co-function of the complement of the angle. <br />
    8. 8. TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES 45Β°,Β 30Β° and 60Β°<br />Β <br />To find the functions of 45Β°, construct an isosceles right triangle with each leg equal to 1, that is, π‘Ž=1 and 𝑏=1.. By Pythagorean Theorem, the hypotenuse 𝑐=2.<br />sin45Β°=12=22<br />cos45Β°=12=22<br />tan45Β°=11=1<br />cot45Β°=11=1<br />sec45Β°=21=2<br />csc45Β°=21=2<br />Β <br />450<br />1<br />450<br />1<br />
    9. 9. To find the functions of 300 and 600, take an equilateral triangle of side 2 and draw the bisector of one of the angles. This bisector divides the equilateral triangle into two congruent right triangles whose angles are 300 and 600. By Pythagorean Theorem the length of the altitude is 3. <br />Β <br />300<br />2<br />600<br />1<br />
    10. 10. sin30Β°=12sin60Β°=32<br />cos30Β°=32cos60Β°=12<br />tan30Β°=13=33tan60Β°=31=3<br />cot30Β°=31=3cot60Β°=13=33<br />sec30Β°=23=233sec60Β°=21=2<br />csc30Β°=21=2csc60Β°=23=233<br />Β <br />
    11. 11. EXAMPLES<br />Draw the right triangle whose sides have the following values, and find the six trigonometric functions of the acute angle A:<br /> a) π‘Ž=8,   𝑏=15<br /> b) 𝑏=21,  𝑐=29<br /> c) π‘Ž=2,  𝑏=3<br />The point (5, 12) is the endpoint of the terminal side of an angle in standard position. Determine the exact value of the six trigonometric functions of the angle.<br />Β <br />
    12. 12. EXAMPLES<br />Find the other five trigonometric functions of the acute angle A, given that:<br /> a) sin𝐴=23<br /> b) sec𝐴=2<br /> c) sin𝐴=2π‘šπ‘›π‘š2+𝑛2<br />Express each of the following in terms of its cofunction:<br /> a) sin73Β°<br /> b) cos20Β°+𝐴<br /> c) cot60Β°βˆ’π›½<br /> d) tan46Β°35β€²23"<br />Β <br />
    13. 13. EXAMPLES<br />Determine the value of  𝛽that will satisfy the ff.:<br /> a) tan2𝛽+10Β°=cot3𝛽<br /> b) sin5π›½βˆ’10=1sec3𝛽+4Β°<br />Evaluate each of the following :<br /> a) sec30Β°βˆ’sin60Β°βˆ’cos30Β°<br /> b) π‘π‘œπ‘‘2Β 45Β°+π‘‘π‘Žπ‘›260Β°βˆ’π‘ π‘–π‘›245Β°<br />Β <br />