TRIGONOMETRY BASIC TRIANGLE STUDY: RATIOS: SINE COSINE TANGENT ANGLES / SIDES SINE LAW: AREA OF A TRIANGLE: - GENERAL TRIGONOMETRY HEROS BASIC TRIANGLE STUDY Complimentary angles: 2 angles = 90 Supplementary angles: 2 angles = 180 Adjacent angles on the same line = 180 Opposite angles on the same line = each other The sum of the interior angles of a triangle = 180 Right triangles have one angle = 90 Pythagorean Theorem = a + b = c RATIOS
SOH CAH TOA TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE B SINE A = = c a COSINE A = = TANGENT A = = C A b The adjacent side is the side next to the reference angle. The opposite side is the side directly across from the reference angle. Remember, it is important to understand that the names of the opposite side and adjacent sides change when you move from one reference angle to the other. RATIOS m A
SIN A COS A TAN A 0 1 0 20 .342 .9397 .364 30 .5
.866 .5774 45 .7071 .7071 1 60 .866 .5 1.7321 80 .9848
.1736 5.6713 90 1 0 0 SIN 30 = COS 30 TAN 30 = = SIN 60 COS TAN 60
SIN 45 COS 45 TAN 45 SINE B SINE A = c a SINE A = C b A COSINE
B COS A = c COS A = a C b A TANGENT B TAN A = c TAN A = a C b A
CALCULATOR CALCULATOR: B The button (key) SIN on the calculator enables you to calculate the value of C SIN A if you know the measurement of ANGLE A. ie. SIN 30 = 0.5 The button (key) on the calculator enables you to calculate the measure of the ANGLE A if you know SIN A ie. () = 30 A ANGLES / SIDES FINDING MISSING SIDES USING TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE,
Finding the measure of x of side BC opposite to the known ANGLE A, knowing the measure of the hypotenuse, requires the use of SIN A. SIN 50 = or x = 5 SIN 50 = 3.83 Finding the measure of y of side AC adjacent to the known ANGLE A, knowing the measure of the hypotenuse, requires the use of COS A. A COS 50 = or y = 5 COS 50 = 3.21 50 5 cm y C B x
ANGLES / SIDES FINDING MISSING SIDES USING TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE (CONTINUED), Finding the measure of x of side BC opposite to the known angle A, knowing also the measure of the adjacent side to angle A, requires the use of TAN A A x TAN 30 = x = 4 TAN 30 = 2.31 cm 4 30 4 cm C B x ANGLES / SIDES
FINDING MISSING ANGLES USING TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE, Finding the acute angle A when its opposite side and the hypotenuse are known values require the use of SIN A. 4 4 ( B SIN A = m A = SIN ) = 53.1 5 5 4 C 5 A ANGLES / SIDES FINDING MISSING ANGLES USING TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE, Finding the acute angle A when its adjacent side and the
hypotenuse are know values require the use of COS A 3 3 COS A = m A = COS = 41.4 B ( ) 4 4 4 C A 3 ANGLES / SIDES FINDING MISSING ANGLES USING TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE, Finding the acute angle A when its opposite side and adjacent side are known values requires the use of TAN A 3 3
B TAN A = m A = TAN = 56.3 ( ) 2 2 3 C A 2 SINE LAW The sides in a triangle are directly proportional to the SINE of the opposite angles to these sides. A c b B C a
The SINE LAW can be used to find the measure of a missing side or angle. CASE 1: Finding a side when we know two angles and a side We calculate the measure of x of AC A 60 B x C 50 15 cm SINE LAW The SINE LAW can be used to find the measure of a missing side or angle. CASE 2: Finding the angle when we know the two sides and the opposite angle to one of these sides We calculate the measure of angle B A 50 B
x 10 cm C 13 cm AREA OF A TRIANGLE GENERAL FORM AREA = HEIGHT or AREA = L W BASE
AREA OF A TRIANGLE TRIGONOMETRIC FORMULA AREA = A AREA = c b h AREA = C B H a
AREA OF A TRIANGLE HEROS FORMULA When you are given the measures for all three sides a, b, c of a triangle, Heros Formula enables you to calculate the area of a triangle. AREA = A b c P = half the perimeter of the triangle C B a A c b
AREA OF A TRIANGLE C a B GENERAL TRIGONOMETRIC 3.55 cm 117.3 8 cm 6 cm 26.4 36.3 12 cm
12 cm A = = 21.3 A= HEROS 8 cm 6 cm A= P= 12 cm A= A = = 21.3 A=