MATH 41 Sections 5.1-5.4 Fundamental Identities Reciprocal Quotient Pythagorean 5 Example: If tanθ = and θ is in quadrant II, find the exact values of the other 1 trigonometric functions using only fundamental identities. Example: Write tan x in terms of sec x.
We have already listed the Co-function Identities when we defined the six trigonometric functions based on the lengths of the sides of a right triangle. Problem: Use a cofunction identity to solve the following equation. Assume all angles invloved are acute angles. ( θ ) = ( θ ) sin 11 cos 3 4 Even and Odd Function Identities Solving trigonometric equations (part I) Example: Solve for the variable in the indicated interval: = for α [ 0 π ) a) tan a seca, (use a calculator and factoring) = for x [ 0 360 ) b) tan x 3 tan x,. (try without a calculator)
Example: Determine all solutions in radians for the equation tan x = 3. (try without a calculator) Example: Determine all solutions in degrees for the equation cosθ + 3 = 0. (try without a calculator) Section 5. Verifying Identities Strategies for verifying Identities Verify the following identity (work on one side of the identity only) cosθ cscθ sinθ secθsinθ = From the Trigonometry review, try problems 17 and 18.
In our textbook, try p 418-419 # 7, 39, and 48 Section 5.3 Sum and Difference Identities for Cosine, Sine, and Tangent Derivation of cos( A B) The remaining sum and difference identities:
Using the Sum and Difference Identities Example: Using identities, find the exact value of each of the following: a) sin 75 π b) cos 1 1 3 Example: Given that cos A = and sin B =, with A in quadrant IV and 13 5 B in quadrant III, find the exact value of the following: a) sin ( A B) b) tan( A+ B) c) cos( A B) d) the quadrant in which A B lies.
Proving a Reduction Formula Verify the following identity (work on one side of the identity only) 3π cos β = sin β Problem: Simplify the difference quotient cos( x+ h) cos x h An Application popular with some calculus professors A problem from calculus is to determine the distance x from a wall in an art gallery an observer must stand in order to get the best view, i.e., maximize the angle subtended at her eye by the painting. Solving this problem requires finding an expression for θ in terms of x. Fill in the steps below to do the trigonometry part of this problem. i) Let x be the distance from the eye of the observer to the wall and let φ be the angle from x to the line of sight from the observer to the bottom of the painting. ii) Suppose h = 4 feet and d = 1.5 feet. Find an expression for tanθ, by using a sum or difference formula for tangent. (hint: Let α = θ + φ ).
iii) Make a general formula for tanθ in terms of d, h, and x. Section 5.4 Multiple Angle Identities Problem: Use the identities cos( A+ B), sin( A+ B), and tan( A+ B) to derive cos( A), sin( A), and tan( A ). Hint: let A = B Example: Use an identity to write each of the following as a single trigonometric function or as a single number. tan15 a) sin( π / 1) cos( π / 1 ) b) tan 15 1
The Half Angle Identites for Sine, Cosine, and Tangent Use a pythagorean identity to find two other forms for the double angle identity for Cosine. The half-angle identities for Sine and Cosine are derived from these other forms! Example: Another problem from calculus is to calculate the expression sin θdθ. In order to accomplish this, sin θ must be written in terms of cos θ. This requires a trigonometric identity. Use an identity to show how this can be done. 1 Problem: A= s sin θ is an area formula for an isosceles triangle where s = a length of one of the equal sides and θ = the measure of an angle opposite s. Derive this area formula.
The Half Angle Identity for Tangent: Example: Use the half angle identity for tangent to find an exact value for tan195 Solving Equations Involving Multiple Angles: Examples: Solve for the variable in the indicated interval a) cost = sin t for x [ 0, 360 ). b) 1 = sin t+ sin t for all t c) cos = 4sin 1 x x for x [ 0 360 ),.
Problem: A graph of y = cos x + cos x for 0 x π is shown below. The x- coordinates of the turning points P, Q and R on the graph are solutions of the equation sin x + sin x = 0. Find the exact values of the coordinates of these points. Application: If a projectile is fired from ground level with an initial velocity of v ft/sec and at an angle of θ degrees with the horizontal, the range R of the projectile is given by R = v 16 sin θ cos θ. If v = 80 ft/sec, approximate the angles that result in a range of 150 feet. Problem: A rectangle is inscribed in a semicircle of radius 1