Name ___________________________ Math 4 February 25, 2015 Unit 4 Trigonometry Review: Concepts & vocabulary Trigonometry – Chapter 4 in text (skipped 4.6 and frequency in applications) Objectives: Use radians fluently in all types of problems where appropriate. Graph sinusoidal functions from an equation, identifying all critical features in the equation as well as on the graph Write an equation of a sinusoidal function given a context or a graph Demonstrate fluency in solving trig problems related to the x-y plane using reference triangles Apply understanding of inverse functions to solve for missing angle measures Apply understanding of functions to correctly limit the domain & ranges as needed, especially in regards to the inverse functions Apply knowledge of transformations to graph trigonometric functions more easily & to be able to compare functions in or out of a context Overview of important vocabulary & skills Radians and degrees – two different units of measure for angles. In right triangle trig everything is always in degrees, working with sinusoidals and graphs, everything is always in radians. The unit circle is the connection between the two different units of measure. If no unit is given, the unit is assumed to be radians. You must use the degree symbol to indicate degrees. Converting from one to the other: 𝜋 Degrees to radians: 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑦 180 Radians to degrees: 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑦 180 𝜋 Arc Length & area of Sectors: find the fraction of the circle you have and multiply it by the circumference for arc length or by the area of the circle for area of a sector. You can also set up a proportion to solve for the missing quantity. Right Triangle Trigonometry: SOHCAHTOA is your friend. You can find any ratio: sine, cosine, tangent or the reciprocal functions: csc, sec, and cot. If you are given 2 lengths, use the Pythagorean theorem to find the 3 side. If you are given one side and an angle you can find a second side. Which then leads you to be able to find the 3rd side (Pythagorean thm again). Inverse Functions in right triangle trig: You can find the angle measure given two sides by writing a trig relationship and then using the inverse function. 6 6 Find theta. 𝑡𝑎𝑛𝜃 = 9 𝑡𝑎𝑛−1 (9) = 𝜃 .588 = 𝜃 or 𝜃 = 33.69° 6 9 Name ___________________________ Math 4 February 25, 2015 Unit 4 Trigonometry Review: Concepts & vocabulary Applications with right triangle Trig: Angles of elevation and depression – using these angles to set up a right triangle, determine the trig function that connects given information and solve for the unknown. Unit Circle: A circle of radius length 1 unit. Gives all the relationships between degrees and radians, and gives trig values of functions for all special triangle values. Clockwise rotations yield negative angles of rotation, counterclockwise rotation yields positive angle rotation. Know the values for all multiples of pi/6; pi/4; and pi/3 and quadrantal angles (multiples of pi/2). Using the unit circle to find values of trig functions: Each point on the circle has coordinates that correspond to it. The x-coordinate is the cosine value of the angle. The y-coordinate is the sine value of the angle. So, to evaluate functions using the circle, you must: 1) find the point that corresponds to the given angle of rotation, 2) find the correct coordinate or if finding tangent take the ratio; 3) you are done. To use the unit circle to help with inverse function problems – rewrite the inverse function so that you are looking for the angle whose sine/cosine/tangent etc. equals the given value. Find that value on the unit circle and your answer is the corresponding angle. Remember the restrictions for angles with inverse functions: Sin-1(x) only quadrants I and IV Tan-1(x) only quadrants I and IV Cos-1(x) only quadrants I and II Sinusoidals – every sine and cosine function is a sinusoidal function. They are periodic and both can be written as a sine function. To graph these you mush identify: Amplitude – half of the distance between the maximum and minimum; the distance from the midline to either the maximum or minimum. In an equation in standard form, it is the coefficient of the sine or cosine function. Remember amplitude is always positive. Midline (also your vertical translation) – in an equation in standard form, it is the constant added to the sine or cosine function; you can find it by taking the average of the maximum and minimum or by subtracting the amplitude from the maximum or adding it to the minimum. Period – the period is the length that it takes to complete one full cycle of the function. In an equation you can find it by dividing 2 by the b-value (coefficient of the x) . Graphically look for one complete cycle, then calculate the distance along the x-axis. Finding b: take 2 and divide it by the period Phase shift (c-value) – the horizontal translation of the function, in the equation in standard form it is the constant added or subtracted from the x-term. Graphically, find the x-value where a period of the function begins. Name ___________________________ Math 4 February 25, 2015 Unit 4 Trigonometry Review: Concepts & vocabulary Reflections: a sine or cosine function can be negative (the negative sign goes before the amplitude). This indicates a reflection over the x-axis prior to vertical translation. Since these functions are periodic, every one of these functions can be written as negative sine or cosine – it just depends on where you choose to start your period. In other words, there are multiple correct equations for any given graph of a sinusoidal. Applications of sinusoidals: find the above information (a, b, c, and d) so that you can write an equation. Often you will be given x-values to evaluate the function at those points – plug in the value for x. Sometimes you will be given the value of the function and are asked to find the xvalue. Set the equation equal to the given number and solve. Reciprocal functions: now the relationships between sine, cosine, and tangent and these functions. Know what the graphs look like generally. Understand domain restrictions and where they come from. Be able to solve for an angle or side in right triangle or unit circle problems. You will not have to graph these, but might need to demonstrate general understanding of the functions. Inverse functions: a key aspect of the inverse functions is that in order to be functions the ranges of the graphs must be restricted. Each function has its own restrictions based on their graphs. These restrictions come into play when solving equations involving the inverse functions.