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Trigonometry is the study of relations of the sides and angles of triangles. In right triangle trigonometry, we will focus on those that involve right angles. We will learn three very important trigonometric ratios based on the Pythagorean Theorem (a.k.a. the trigonometry right triangle formula): Sine, Cosine, and Tangent.
Pythagorean Theorem is summarized by the formula \( a^2+b^2=c^2 \), where a and b are the sides of the right triangle, and c is the hypotenuse (the longest side of the right triangle). For the two sides which are not the hypotenuse, we name them either opposite or adjacent, depending on which angle we are looking at in the calculation. It might sound confusing, but no worries, we have a mnemonic device to help us memorize these angle-side relationships — SohCahToa.
We first name the angle that we focus on as Theta (θ). Then, we can name the side that is opposite to angle θ as “opposite” and the side that is next to angle θ as “adjacent”. Depending on which angle and side we know, we find the missing side length of a right triangle in this way:
As long as we know the measures of any two sides, or an angle and a side of a right triangle, we are able to use the three trig ratios to answer questions like how to find hypotenuse, how to find the missing side of a triangle, and how to find the measure of an angle.
We will also look at two types of special triangles: 30-60-90 triangles and 45-45-90 triangles. They are right triangles with angles of 30°-60°-90° and 45°-45°-90°. They are called special triangles because their trig ratios give exact values that are very easy to memorize. This is nice because we won’t need to use a calculator to evaluate the trig ratios.
Triangles, angles, and sides are the basic foundation of trigonometry. In trigonometric ratios and angle measure we will brush up on these parts once more. You will learn about the different types of angles such as: standard angles, reference angles, and co-terminal angles. You will also learn how to apply the ASTC rule to determine if a trig ratio is positive or negative.
To further model the relationships between angles and sides of a triangle, be it a right-angled triangle, an obtuse triangle, or an acute triangle, we will look at the law of sines and law of cosines. We will also learn the conversion between degrees and radians, the two units of measuring angles, and how they can be used to find the length of an arc.
And lastly, we also introduce another very important concept that is used extensively in the realm of trigonometry, the unit circle.
The unit circle, is a circle with a radius of one. It is important because it allows us to determine the sines and cosines of any angle θ in right triangles as long as we are dealing with real numbers. Once we know the trick to memorizing the unit circle, filling out the unit circle chart is just a piece of cake.
Once we have an understanding of the trigonometry basics, including terminology, trig ratios (SohCahToa), angle measurement, the ASTC rule, and the unit circle. From the unit circle lessons, we have learned that, for a point on the unit circle, its x-coordinate is equal to cos θ and y-coordinate is equal to sin θ. We will use these basic concepts and learn how to graph the movements of this point as it moves along the unit circle with respect to different trigonometric functions.
The trig graphs that we will learn to draw are: sine graph, cosine graph, tangent graph, cotangent graph, secant graph and cosecant graph. To draw these graphs, we will use angles measured in radians as our x value, and plug them into the trig function to find the y values. Then, we plot these ordered pairs out on an x-y plane to get the graphs.
After learning how to graph the six trigonometric functions, we will look into how the graphs can be transformed. A transformation can change the graph of a trig function in a couple ways: vertical expansion/contraction, horizontal expansion/contraction, and vertical/horizontal shifts. We can easily identify any changes of the trig functions by graphing the trig functions for multiple periods; stating the vertical displacement, phase shift, period and amplitude; and also finding the domain and range of the transformed functions. We will also try to find the equation of a trigonometric function from its graph.
Trigonometry can be confusing for some students, especially if they don’t know how to apply them to real life applications. Now that we have explored the six trigonometric functions and their trigonometric graphs, if we were to apply some transformations, we can use these functions to make models in predicting certain outcomes or even a pattern of a phenomenon with respect to time. So, let’s look at some real world applications of the trigonometric functions.
One example of an application of trigonometric functions would be of the most common theme park ride — the Ferris wheel. In fact, we can model the distance between a person riding on the Ferris wheel to the ground in relation to time using a sinusoidal function. To solve this Ferris wheel trig problem, we will need to make use of the radius of the Ferris wheel, the time it takes for each revolution and the lowest height of the person’s seat from the ground at time zero.
We can also use the trigonometric function to find out the best time to go surfing too. As long as we know the time between a minimum depth and a maximum depth of the water, we can model the low tide, high tide and water depth over time with a sinusoidal function as well.
Another example is to model the simple harmonic motion of springs over time. Applying the same concept used in solving the Ferris wheel and tidal wave trig problems, we are able to use trig functions to find the pattern of the motions of a spring versus time.
There are also a lot of applications of the trigonometric functions in astronomy, geometry and navigation. The more you try to understand the concepts, the more you will see how trigonometry is everywhere around you.
As we deal with more and more advanced trigonometry problems, only knowing the trigonometric ratios and the graphs of sine, cosine, tangent and their reciprocals will not be enough. This is when we need trigonometric identities. Identities in mathematics are equations which always hold true for any values of the variables. Among them, trig identities are identities that involve trig functions. Identities are important and convenient because any member in an identity can be replaced by another and the identity will still hold true. For example, we can replace sec θ with θ, and vice versa.
Same as the basic trig ratios we have learned in previous chapters, SohCahToa (sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, and tangent = opposite/adjacent), the identities we will learn in this chapter are also derived from the Pythagorean Theorem. We will focus on six different trig identities: Quotient identities, Reciprocal identities, Pythagorean identities, Sum and difference identities/sum and difference formula, Cofunction identities, and Double-angle identities/double-angle formula. We will try to prove these trig formulas. Then, we will learn how to use them to simplify trigonometric expressions and evaluate trigonometric equations.
You can also download our Trigonometric Identities chart as a handy PDF for future reference.
Trigonometric equations are equations that have one or more of the trigonometric functions. These trig functions can be sine, cosine, tangent, cotangent, secant or cosecant.
In the previous chapters, we have learned about trigonometric ratios, trigonometric functions, and trigonometric identities. By now, we already have a clear grasp of trigonometry. So, in this chapter, we will try to learn a bit more by applying all these concepts we have learned to solve trig equations. It will be very useful to review our lessons on trigonometric functions, unit circle, reference angles, co-terminal angles, and standard angles to effectively understand this chapter.
We will learn how to solve first-degree, second-degree trigonometric equations, and also trig equations involving multiple angles. We will look at how to solve for theta (or x) of an equation, whether it is measured in radian or degree. Yet, we need to be extra careful when we come across trig expressions in the form of a fraction. As we all know very well, in a fraction, the denominator cannot be zero. If it is equal to 0, then the fraction is undefined. By the same token, when we are solving trig functions in fraction form, we need to find the non-permissible values for the trig expressions before we can solve for any missing angles. When a trig equation involves only one type of trigonometric function, we can easily solve it by factoring. However, for trig equations that involve more than one type of trig functions, before we can go ahead and solve them, we will need to use the trig identities to convert the trig functions into one type. In this chapter, we will learn how to use Pythagorean identities, sum and difference identities, and double-angle identities to solve for trig equations. It’s always a good idea to have the trigonometric identities and formulas next to you when going through this chapter.
By now, we should be very familiar with six trigonometric ratios: sine, cosine, tangent, cosine, cosecant, and cotangent. However, these trig ratios can only help us find the length of a right triangle when we are given the measures of an angle and a side. What if we are given only the length of two sides of a right triangle and are asked to look for a missing angle? This is where the inverse trig functions come in. As its name suggests, inverse trig functions are essentially the opposite of sine, cosine and tangent. For example, we know that
In this chapter, we will start off with looking at the graphs of the inverse trig functions and their domain and range. Then, based on the cancellation laws derived from the graphs, we will learn how to solve inverse trigonometric functions. Inverse trig functions are just like any other functions, and so they also have corresponding reciprocal functions. We will try to derive their graphs, find their domain and range, and find the exact values of some inverse reciprocal trig functions.
When we learned about square and square roots, we were taught that we cannot take the square root of a negative number because the square of any real number is always positive. However, there are situations when we do need to take the square root of a negative number to complete our calculation. Therefore, for the ease of calculation, an imaginary number, i, was invented. i is defined as the square root of negative 1
After going through this chapter, we will see how imaginary and complex numbers are used extensively in the field of trigonometry. Before we dive into any applications, we will first learn all the basics about imaginary and complex numbers. After we’ve got a good grasp of adding, subtracting, multiplying, and dividing complex numbers, we will try to find the distance and midpoint of two complex numbers using the distance formula. The calculation is very similar to how we find the distance and midpoint of two points on an x-y plane. Then, we need to make use of our knowledge on the Pythagorean Theorem and SohCahToa to find the angle and absolute value of complex numbers. Lastly, we will talk about the conversion between rectangular form and polar form and the operations in polar form, which some of them require the use of the unique characteristics of special right triangles.