Right Triangle Trigonometry Trigonometry is based upon ratios of the sides of right triangles. The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. θ hyp opp adj The sides of the right triangle are:  the side opposite the acute angle ,  the side adjacent to the acute angle ,  and the hypotenuse of the right triangle.

Right Triangle Trigonometry The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides refer to another angle, other than the 90o. A

S O H C A H T O A Trigonometric Ratios hyp opp θ adj The trigonometric functions are: sine, cosine, tangent, cotangent, secant, and cosecant. sin  = cos  = tan  = csc  = sec  = cot  = opp hyp adj S O H C A H T O A

Reciprocal Functions sin  = 1/csc  csc  = 1/sin  cos  = 1/sec  sec  = 1/cos  tan  = 1/cot  cot  = 1/tan 

Finding the ratios The simplest form of question is finding the decimal value of the ratio of a given angle. Find using calculator: sin 30 = sin 30 = cos 23 = tan 78 = tan 27 = sin 68 =

Using ratios to find angles It can also be used in reverse, finding an angle from a ratio. To do this we use the sin-1, cos-1 and tan-1 function keys. Example: sin x = 0.1115 find angle x. sin-1 0.1115 = shift sin ( ) x = sin-1 (0.1115) x = 6.4o 2. cos x = 0.8988 find angle x cos-1 0.8988 = shift cos ( ) x = cos-1 (0.8988) x = 26o

Calculate the trigonometric functions for  . 4 3 5  The six trig ratios are sin  = sin α = cos  = cos α = tan  = What is the relationship of α and θ? tan α = cot  = cot α = sec  = They are complementary (α = 90 – θ) sec α = csc  = csc α =

Cofunctions sin  = cos (90  ) cos  = sin (90  ) tan  = cot (90  ) cot  = tan (90  ) tan  = cot (π/2  ) cot  = tan (π/2  ) sec  = csc (90  ) csc  = sec (90  ) sec  = csc (π/2  ) csc  = sec (π/2  )

Finding an angle from a triangle To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle. We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio. 14 cm 6 cm C 1. Find angle C Identify/label the names of the sides. b) Choose the ratio that contains BOTH of the letters.

14 cm 6 cm C 1. We have been given the adjacent and hypotenuse so we use COSINE: Cos A = h a Cos A = Cos C = Cos C = 0.4286 C = cos-1 (0.4286) C = 64.6o

Find angle x 2. 8 cm 3 cm x Given adj and opp need to use tan: Tan A = a o Tan A = Tan x = Tan x = 2.6667 x = tan-1 (2.6667) x = 69.4o

3. 12 cm 10 cm y Given opp and hyp need to use sin: Sin A = sin A = sin x = sin x = 0.8333 x = sin-1 (0.8333) x = 56.4o

Finding a side from a triangle 7 cm k 30o 4. We have been given the adj and hyp so we use COSINE: Cos A = Cos A = Cos 30 = Cos 30 x 7 = k 6.1 cm = k

4 cm r 50o 5. We have been given the opp and adj so we use TAN: Tan A = Tan A = Tan 50 = Tan 50 x 4 = r 4.8 cm = r

12 cm k 25o 6. We have been given the opp and hyp so we use SINE: Sin A = sin A = sin 25 = Sin 25 x 12 = k 5.1 cm = k

x = x 5 cm 30o 1. Cos A = Cos 30 = x = 5.8 cm 4 cm r 50o 2. Tan 50 x 4 = r 4.8 cm = r Tan A = Tan 50 = 3. 12 cm 10 cm y y = sin-1 (0.8333) y = 56.4o sin A = sin y = sin y = 0.8333

Example: Given sec  = 4, find the values of the other five trigonometric functions of  . Solution: Draw a right triangle with an angle  such that 4 = sec  = = . adj hyp θ 4 1 Use the Pythagorean Theorem to solve for the third side of the triangle. sin  = csc  = = cos  = sec  = = 4 tan  = = cot  =

Applications Involving Right Triangles A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.3. How tall is the Washington Monument? Solution: where x = 115 and y is the height of the monument. So, the height of the Washington Monument is y = x tan 78.3  115(4.82882)  555 feet.

Fundamental Trigonometric Identities Reciprocal Identities sin  = 1/csc  cos  = 1/sec  tan  = 1/cot  cot  = 1/tan  sec  = 1/cos  csc  = 1/sin  Co function Identities sin  = cos(90  ) cos  = sin(90  ) sin  = cos (π/2  ) cos  = sin (π/2  ) tan  = cot(90  ) cot  = tan(90  ) tan  = cot (π/2  ) cot  = tan (π/2  ) sec  = csc(90  ) csc  = sec(90  ) sec  = csc (π/2  ) csc  = sec (π/2  ) Quotient Identities tan  = sin  /cos  cot  = cos  /sin  Pythagorean Identities sin2  + cos2  = 1 tan2  + 1 = sec2  cot2  + 1 = csc2 