Pre­Calculus Notes 4.3 Right Triangle Trigonometry 6 Ratios of sides of a right triangle and their names: hypotenuse c b opposite θ a adjacent The terms sine, cosine, tangent, secant, cosecant, and cotangent are the names given to each of the combinations of the ratios of the sides. Use SOH­CAH­TOA to determine the 6 trig ratios. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ = Notice the reciprocal relationships:
csc q =
1 sin q
sec q =
1 cos q
cot q = 1 tan q
Example 1: Use the triangle to find the ratios of all six trig functions. (Leave answers as reduced improper fractions.) B 13 12 A q C
Exact values of trig ratios can be determined for all angles that are multiples of 30º or 45º by determining a point P(a,b) on the terminal side of the angle and sketching the reference triangle for the angle. We can use the ratio of sides of the familiar 30­60­90 degree triangle or the 45­45­90 degree triangle. Sketch the special triangles here and label the ratio of sides. Example 2: Find the value of the missing sides. a.) 45° 6 b.) c.) 60° 7 7 9 Example 3: Use your knowledge of special triangles to determine the values of: a.) sin 30º p b.) cos 4
c.) tan 60º. p d.) cot 6
p e.) sec 4
p
f.) csc 3
Cofunctions of complementary angles are . If q is an acute angle, then:
sin( 90 ° - q ) =
cos(90 ° - q ) =
tan(90 ° - q ) =
cot(90 ° - q ) = ___________
sec(90 ° - q ) =
______
q
csc(90 ° - q ) =
Example 4: Find each in terms of another trig function. a.) sin 28 o = ___________ b.) cot 47 o = ___________ c.) sec81 o = ___________ Example 5: Sketch a right triangle corresponding to the trigonometric function of the acute angle q. Use the Pythagorean Theorem to determine the third side of the triangle and then find the other five trig ratios. 5 a.) cos q = 12
b.) csc q = 4
4.3 Continued Day 2: Calculator and Identities Evaluating trig ratios with the graphing calculator: MODE settings: Set your calculator to degree mode only when determining trig ratios for angles in degree measure. Set your calculator to radian mode when determining trig ratios for angles in radian measure or for real numbers. 1 Use reciprocal relationships to find csc θ, sec θ, and cot θ. (Caution: sin - 1 q ¹
) sin q
Example 1: Evaluate to 4 significant digits. a. cos 38.27° p c. cot 7
b. csc 77°53’ Find the angle given the trig ratio: Example 2: Given sin θ = .4010, find θ in degrees and in radians. Other ways to write sin q = 0.4010: sin ­1 (0.4010) = q OR arcsin (0.4010) = q
d. tan 1.003 IDENTITIES Reciprocal Identities csc x = 1 sin x
sec x = 1 cot x = cos x
1 tan x
Quotient Identities tan x = sin x cos x
cot x = cos x sin x
Pythagorean Identities 2
2 sin x + cos x = 1 2
2 2
tan x + 1 = sec x
2 1 + cot x = csc x
Use for Identities: A. Simplifying. 2 sin q
1. + 1 2 cos q
2. tan x - cot x tan x + cot x
B. Verifying the Identity. Simplify one side until it looks like the other side. 1. sin x csc x = 1 2. (sec q + tan q )(sec q - tan q ) = 1
3. (sec x - 1)(sec x + 1) = tan 2 x
4. tan x + cot x = csc 2 x tan x
4.3 Continued Day 3: Applications of Right Triangles Angle of Elevation – the acute angle that is formed from the horizontal looking up. Angle of Depression – the acute angle that is formed from the horizontal looking down. Use trig ratios and /or Pythagorean Theorem to solve for the unknown. Example 1. If the sun is 30° up from the horizon and shining on a tree forming a 50­foot shadow, how tall is the tree? Example 2. If a rope tied to the top of a flagpole is 35 feet long, then what angle is formed by the rope and the ground when the rope is pulled to the ground, 25 feet from the base of the pole? Example 3: The angle of elevation from a point on the ground to the top of a mountain is 25°. After continuing along a straight path towards then base of the mountain for 1.0 km, another reading is taken. Now the angle of elevation is 42°. Find the height of the mountain.