MTH 112: Elementary Functions

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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.

1 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

2 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

3 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 = = 144 = 12. sin(θ) = hyp = 5 adj, cos(θ) = 13 csc(θ) = hyp =?, hyp = hyp sec(θ) = adj =?,, tan(θ) = adj =?, adj cot(θ) = =?

4 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 ) x : 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).

5 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) undefined 1 undefined 30 π π π π 1 0 undefined 0 undefined : 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.

2: Right triangle trigonometry 12/16 Angle of elevation and angle of depression Many times in application examples involving trigonometry, we consider

6 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 = : 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.

7 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 How tall is the tree? : Right triangle trigonometry 14/16 h tan(61.5 ) = h 200 h = 200tan(61.5 ) =

8 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 : Right triangle trigonometry 16/16 Example 12 d 500 tan(12 ) = 500 d d = 500 tan(12 ) =

Mathematics for Computer Graphics. Trigonometry

Mathematics for Computer Graphics Trigonometry Trigonometry...????? The word trigonometry is derived from the ancient Greek language and means measurement of triangles. trigonon triangle + metron measure