Right Triangle Trigonometry

Objectives Find trigonometric ratios using right triangles. Use trigonometric ratios to find angle measures in right triangles.

History Right triangle trigonometry is the study of the relationship between the sides and angles of right triangles. These relationships can be used to make indirect measurements like those using similar triangles.

Only Apply to Right Triangles Trigonometric Ratios Only Apply to Right Triangles

The 3 Trigonometric Ratios The 3 ratios are Sine, Cosine and Tangent

The Amazing Legend of… Chief SohCahToa

The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are: θ hyp  the side opposite the acute angle , opp  the side adjacent to the acute angle ,  and the hypotenuse of the right triangle. adj The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. sin 𝜃= 𝑜𝑝𝑝 ℎ𝑦𝑝 cos 𝜃= 𝑎𝑑𝑗 ℎ𝑦𝑝 tan 𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗 csc 𝜃= ℎ𝑦𝑝 𝑜𝑝𝑝 sec 𝜃= ℎ𝑦𝑝 𝑎𝑑𝑗 cot 𝜃= 𝑎𝑑𝑗 𝑜𝑝𝑝

EVALUATING TRIGONOMETRIC FUNCTIONS 𝑎=4 𝑎𝑛𝑑 𝑏=3 What is the value of h? Find all six trig functions of angle A sin 𝜃= 𝑜𝑝𝑝 ℎ𝑦𝑝 4 5 csc 𝜃= ℎ𝑦𝑝 𝑜𝑝𝑝 5 4 𝟓 =𝟒 cos 𝜃= 𝑎𝑑𝑗 ℎ𝑦𝑝 3 5 sec 𝜃= ℎ𝑦𝑝 𝑎𝑑𝑗 5 3 cot 𝜃= 𝑎𝑑𝑗 𝑜𝑝𝑝 tan 𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗 4 3 3 4 =𝟑 Remember SOH CAH TOA and the reciprocal identities

EVALUATING TRIGONOMETRIC FUNCTIONS 𝑎=12 𝑎𝑛𝑑 𝑏=5 What is the value of h? Find all six trig functions of angle A sin 𝜃= 𝑜𝑝𝑝 ℎ𝑦𝑝 12 13 csc 𝜃= ℎ𝑦𝑝 𝑜𝑝𝑝 13 12 𝟏𝟑 =𝟏𝟐 cos 𝜃= 𝑎𝑑𝑗 ℎ𝑦𝑝 5 13 sec 𝜃= ℎ𝑦𝑝 𝑎𝑑𝑗 13 5 cot 𝜃= 𝑎𝑑𝑗 𝑜𝑝𝑝 tan 𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗 12 5 5 12 =𝟓 Remember SOH CAH TOA and the reciprocal identities

EVALUATING TRIGONOMETRIC FUNCTIONS 𝑎=1 𝑎𝑛𝑑 ℎ=3 What is the value of b? Find all six trig functions of angle A sin 𝜃= 𝑜𝑝𝑝 ℎ𝑦𝑝 1 3 csc 𝜃= ℎ𝑦𝑝 𝑜𝑝𝑝 3 𝟑 =𝟏 cos 𝜃= 𝑎𝑑𝑗 ℎ𝑦𝑝 2 2 3 sec 𝜃= ℎ𝑦𝑝 𝑎𝑑𝑗 3 2 4 cot 𝜃= 𝑎𝑑𝑗 𝑜𝑝𝑝 tan 𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗 2 4 2 2 =𝟐 𝟐 Remember SOH CAH TOA and the reciprocal identities

Calculate the trigonometric functions for a 45 angle. 1 45 sin 45 = = = cos 45 = = = hyp adj tan 45 = = = 1 adj opp cot 45 = = = 1 opp adj sec 45 = = = adj hyp csc 45 = = = opp hyp

Geometry of the 30-60-90 triangle 2 Consider an equilateral triangle with each side of length 2. 60○ 30○ 30○ The three sides are equal, so the angles are equal; each is 60. The perpendicular bisector of the base bisects the opposite angle. 1 1 Use the Pythagorean Theorem to find the length of the altitude, 3 .

Calculate the trigonometric functions for a 30 angle. 1 2 30 sin 30°= 𝑜𝑝𝑝 ℎ𝑦𝑝 = 1 2 csc 30°= ℎ𝑦𝑝 𝑜𝑝𝑝 = 2 1 =2 cos 30°= 𝑎𝑑𝑗 ℎ𝑦𝑝 = 3 2 sec 30°= ℎ𝑦𝑝 𝑎𝑑𝑗 = 2 3 = 2 3 3 tan 30°= 𝑜𝑝𝑝 𝑎𝑑𝑗 = 1 3 = 3 3 cot 30°= 𝑎𝑑𝑗 ℎ𝑦𝑝 = 3 1 = 3

Calculate the trigonometric functions for a 60 angle. 1 2 60 sin 60°= 𝑜𝑝𝑝 ℎ𝑦𝑝 = 3 2 csc 60°= ℎ𝑦𝑝 𝑜𝑝𝑝 = 2 3 = 2 3 3 cos 60°= 𝑎𝑑𝑗 ℎ𝑦𝑝 = 1 2 sec 60°= ℎ𝑦𝑝 𝑎𝑑𝑗 = 2 1 =2 cot 60°= 𝑎𝑑𝑗 ℎ𝑦𝑝 = 1 3 = 3 3 tan 60°= 𝑜𝑝𝑝 𝑎𝑑𝑗 = 3 1 = 3

TRIG FUNCTIONS & COMPLEMENTS Two positive angles are complements if the sum of their measures is 90° 𝑜𝑟 𝜋 2 . Example: 70° 𝑎𝑛𝑑 20° are complement because 70°+20°=90°. The sum of the measures of the angles in a triangle is 180°. In a right triangle, we have a 90° angle. That means that the sum of the other two angles is 90°. Those two angles are acute and complement. If the degree measure of one acute angle is 𝜃, then the degree measure of the other angle is (90°−𝜃).

TRIG FUNCTIONS & COMPLEMENTS Compare sin 𝜃 and cos (90°−𝜃) . sin 𝜃= 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝜃 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝑦 𝑟 cos 90°−𝜃 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 (90°−𝜃) 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝑦 𝑟 Therefore, sin 𝜃= cos (90°−𝜃) . If two angles are complements, the sine of one equals the cosine of the other.

sin 𝜃=cos⁡( 𝜋 2 −𝜃 ) cos 𝜃=sin⁡( 𝜋 2 −𝜃 ) tan 𝜃=cot⁡( 𝜋 2 −𝜃 ) cot 𝜃=tan⁡( 𝜋 2 −𝜃 ) sec 𝜃=csc⁡( 𝜋 2 −𝜃 ) csc 𝜃=𝑠𝑒𝑐( 𝜋 2 −𝜃 )

Using cofunction identities Find a cofunction with the same value as the given expression: sin 72° sin 72°= cos 90°−72° = cos 18° csc 𝜋 3 csc 𝜋 3 = sec ( 3𝜋 6 − 2𝜋 6 )= sec 𝜋 6 =sec⁡( 𝜋 2 − 𝜋 3 )

Angle of Elevation

Angle of Depression

Angle of ELEVATION AND DEPRESSION

A surveyor is standing 50 feet from the base of a large tree A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree? tan 71.5° ? tan 71.5° 71.5° y = 50 (tan 71.5°) 50 y = 50 (2.98868) Look at the given info. What trig function can we use?

A person is 200 yards from a river A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge? cos 60° x (cos 60°) = 200 200 60° x x X = 400 yards Look at the given information. Which trig function should we use?

h = (13.74 + 2) meters x h = 15.74 meters A guy wire from a point 2 m from the top of an electric post makes an angle of 700 with the ground. If the guy wire is anchored 5 m from the base of the post, how high is the pole? 2 m h = (13.74 + 2) meters Guy wire x 700 h = 15.74 meters 5 m Which trig function should we use?

Great job, you guys!