6.2: Right triangle trigonometry 1/16 Figure: Euclid of Alexandria was a Greek mathematician, often referred to as the Father of Geometry 6.2: Right triangle trigonometry 2/16 6.2: Right triangle trigonometry Standard labeling for any triangle C b γ a A α c β B Vertices: A, B and C Angles: α, β and γ Sides: a,b and c a is osite to α b is osite to β c is osite to γ
6.2: Right triangle trigonometry 3/16 Similar triangles C G F A E D B Similar triangles Similar triangles have congruent corresponding angles. Corresponding sides of similar triangles are proportional They are NOT necessarily the same size 6.2: Right triangle trigonometry 4/16 C osite hypotenuse A adjacent Six possible side ratios: θ B hyp, adj hyp, adj, hyp, hyp adj, adj
6.2: Right triangle trigonometry 5/16 Right triangle definitions of trigonometric functions Let θ be the acute angle of a right triangle. C hypotenuse osite A θ adjacent B sin(θ) = hyp, csc(θ) = hyp, adj cos(θ) = hyp, hyp sec(θ) = adj, tan(θ) = adj, adj cot(θ) = 6.2: Right triangle trigonometry 6/16 Example Example 1 Find the six trigonometric functions for the angle θ, where the length of the osite side is 5 and the length of the hypotenuse is 13. Solution: By the Pythagorean theorem, the length of the side adjacent to θ is: adjacent = 13 2 5 2 = 144 = 12. sin(θ) = hyp = 5 adj, cos(θ) = 13 csc(θ) = hyp =?, hyp = 12 13 hyp sec(θ) = adj =?,, tan(θ) = adj =?, adj cot(θ) = =?
6.2: Right triangle trigonometry 7/16 Reciprocal identities (or ratio identities) csc(θ) = 1 sin(θ), sec(θ) = 1 cos(θ), cot(θ) = 1 tan(θ). Example 2 Find the exact value of: cot(45 ), sec(30 ) and csc(60 ). 30 30 45 2 x 2 1 2 60 60 45 1 1 1 6.2: Right triangle trigonometry 8/16 Cofunction identities c α b Write the values of each trigonometric function for the angles α and β : sin(α) = a c = cos(β) β a tan(α) = a b = cot(β) sec(α) = c b = csc(β).
6.2: Right triangle trigonometry 9/16 Some exact values of trigonometric functions Degrees Radians Trigonometry Function Table sin(θ) cos(θ) tan(θ) cot(θ) sec(θ) csc(θ) 0 0 0 1 0 undefined 1 undefined 30 π 1 3 3 2 3 3 2 6 2 2 3 3 45 π 2 2 1 1 2 2 4 2 2 60 π 3 1 3 2 3 3 2 3 2 2 3 3 90 π 1 0 undefined 0 undefined 1 2 6.2: Right triangle trigonometry 10/16 α and β are complementary angles α + β = 90 sin(α) = cos(90 α) = cos(β), tan(α) = cot(90 α) = cot(β), sec(α) = csc(90 α) = csc(β), cos(α) = sin(90 α) = sin(β), cot(α) = tan(90 α) = tan(β), csc(α) = sec(90 α) = sec(β). Cofunctions of complementary angles are equal.
6.2: Right triangle trigonometry 11/16 Example 3 It is known that tan ( π 8 ) = 2 1. Find tan ( 3π8 ). ( ) 3π ( π tan = tan 8 2 π ) 8 ( π ) = cot 8 = 1 tan ( ) π = 1 = 2 + 1 8 2 1 6.2: Right triangle trigonometry 12/16 Angle of elevation and angle of depression Many times in application examples involving trigonometry, we consider angles formed by a horizontal line and the line of sight from a reference point on the horizontal line to an object below or above it. We refer to such an angle as: Angle of elevation if the object is above the horizontal. Angle of depression if the object is below the horizontal.
6.2: Right triangle trigonometry 13/16 Some applications Example 4: Hight of the world tallest tree According to the The Guinness Book of World Records, the tallest tree currently standing is the Mendocino Tree, a coast redwood at Montgomery State Reserve near Ukiah, California. At a distance of 200 feet from the base of the tree, the angle of elevation to the 61.5 top of the tree is approximately 61.5. How tall is the tree? 200 6.2: Right triangle trigonometry 14/16 h tan(61.5 ) = h 200 h = 200tan(61.5 ) = 200 1.8417 368.35 200 61.5
6.2: Right triangle trigonometry 15/16 Example Example 5: Pinpointing a Fire A U.S. Forest Service helicopter is flying at a height of 500 feet. The pilot spots a fire in the distance with an angle of depression of 12. Find the horizontal distance to the fire. 12 500 6.2: Right triangle trigonometry 16/16 Example 12 d 500 tan(12 ) = 500 d d = 500 tan(12 ) = 500 0.2125 2352.315