8.1 Simple Trig Equations

There are often multiple (infinite) solutions to trigonometric equations. For example take the equation sin(x)=.5. Find the solutions.

With the sine and cosine, because the period is 2pi, or the distance that it takes to repeat itself is 2pi, every time you find a solution if you add a multiple of 2pi to that solution you will find another solution.

Solve this equation just like you normally would, trying to isolate the variable, understand that the variable is stuck to cosine so you are actually going to isolate cos ϴ. Now ask yourself, where is cos ϴ equal to -1. Its at π, and then at 3π, and then 5π So solution is π+2πk

This is a little trickier, we know that the cosine is negative so it exists in the 2 nd and 3 rd quadrants. But first we must find a reference angle that exists in the 1 st quadrant, so neglect the negative and find cos -1 (2/3). That will yield This is the 1 st quadrant reference angle. Now take that reference angle and place it in the 2 nd and 3 rd quadrants radians or ⁰ Now looking for these 2 angles π In order to do this, take and add it to pi to find the yellow angle, and subtract it from pi to find the red one.

The same idea can be thought of using the graph of cos(ϴ) 48.2⁰ 131.8⁰228.2⁰

Slopes of lines with the use of an angle of inclination. If we look at slope as rise/run we end up with the idea of a right triangle. y = x α rise run Rise is opposite alpha. Run is adjacent to alpha. So if we talk about the angle alpha we recognize the relationship of tangent. Thus tan(α)=rise/run Thus tan(α) = m And α = tan -1 (m) An angle of inclination is the angle formed by a line and the horizontal (or in this case the x axis).

m = tan(α) and α = tan -1 (m) Line l passes through the point (-1,3) and makes an angle of 70⁰ with the x-axis, find its slope to the nearest hundredths. Then find the equation for line l.

Line l has an equation of 9x+4y=108. First where does the line cross the x-axis? What is the angle of inclination for the line?

Consider the two lines. l 1  5x+3y=30 l 2  5x-2y=-10 Find the measure of the acute angle alpha that they form at their intersection. α

Homework pg evens 19-24, 27, 28