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Trigonometry Basics

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How to Calculate the Sine of Special Angles in Degrees

Measuring angles in degrees for the sum and difference formulas for sine is easier than measuring in radians, because adding and subtracting degrees is much easier than adding and subtracting radians. [more…]

How to Calculate the Sine of Special Angles in Radians

You can use the concept of sum and difference formulas to calculate the sine of special angles in radians. This process is different than solving equations because here you're asked to find the trig value [more…]

How to Apply the Sine Sum and Difference Formulas to Trig Proofs

When dealing with sine sum and difference formulas in a trig proof, you need to make one side of the given equation look like the other. You can work on both sides to get a little further if need be, but [more…]

Trigonometry Proofs and Pythagorean Identities

The Pythagorean identities pop up frequently in trig proofs. Pay attention and look for trig functions being squared. Try changing them to a Pythagorean identity and see whether anything interesting happens [more…]

How to Find Trigonometric Functions of an Angle by Using Pythagorean Identities

You can use Pythagorean identities to find the trig function of an angle if you know one trig function of the angle and are looking for another. For example, if you know the sine of an angle, you can use [more…]

How to Solve a Triangle When You Know Two Consecutive Side Lengths (SSA)

In some trig problems, you may be given two sides of a triangle and an angle that isn't between them, which is the classic case of SSA, or Side-Side-Angle [more…]

How to Solve a Triangle When You Know Two Angle Measures

If you know two angle measures and a side length on a triangle, you can use the Law of Sines to find the missing parts of the triangle. In this case, you need to know either two angles and the side in [more…]

How to Combine Transformations with a Sine or Cosine Graph

Sometimes you will be asked to graph a sine or cosine function with more than one transformation. For example, you may need to change the amplitude of the graph as well as shift it horizontally. When performing [more…]

How to Prove Trigonometric Identities When Terms Are Being Added or Subtracted

When the terms in a trig proof are being added or subtracted, you may create fractions where none were before. This is especially true when dealing with secant and cosecant, because you create fractions [more…]

How to Work with 45-45-90-Degree Triangles

All 45-45-90-degree triangles (also known as 45ers) have sides that are in a unique ratio. The two legs are the exact same length, and the hypotenuse is that length times the square root of 2. The figure [more…]

How to Apply the Sum and Difference Formulas for Tangent to Trig Proofs

The sum and difference formulas for tangent are very useful if you want to prove some basic trig identities. For example, you can prove the co-function identities by using the difference formula and the [more…]

How to Calculate Values for the Six Trigonometric Functions

In pre-calculus, you need to evaluate the six trig functions — sine, cosine, tangent, cosecant, secant, and cotangent — for a single angle on the unit circle. For each angle on the unit circle, three other [more…]

How to Use a Reference Angle to Find Solution Angles

In pre-calculus, you use trig functions to solve algebraic equations. When you find the value of the angle in an equation, which is the angle that is a solution to the equation, you use that as the [more…]

How to Graph a Cosecant Function

Cosecant is almost exactly the same as secant because it's the reciprocal of sine (as opposed to cosine). Anywhere sine has a value of 0, you see an asymptote in the cosecant graph. Because the sine graph [more…]

How to Prove an Equality Using Pythagorean Identities

When asked to prove an identity, if you see a negative of a variable inside a trig function, you automatically use an even/odd identity. You first replace all trig functions with a negative variable inside [more…]

How to Prove an Equality Using Co-function Identities

Co-function identities may pop up in trig proofs. If you see the expression pi/2 – x in parentheses inside any trig function, you need to use a co-function identity for the proof. Follow the steps to prove [more…]

How to Simplify an Expression Using Periodicity Identities

Periodicity identities illustrate how shifting the graph of a trig function by one period to the left or right results in the same function. The functions of sine, cosine, secant, and cosecant repeat every [more…]

Applying the Sum and Difference Formulas for Cosines to Find the Cosine of the Sum or Difference of Two Angles

You can use the sum and difference formulas for cosine to calculate the cosine of the sums and differences of angles similarly to the way you can use the sum and difference formulas for sine, because the [more…]

How to Apply the Sum and Difference Formulas for Cosine to Trig Proofs

You can use the sum and difference formulas for cosine to prove trig identities. When working with sum and difference formulas for cosines, you're simply plugging in given values for variables. Just make [more…]

How to Find the Tangent of the Sum or Difference of Angles

As with sine and cosine, you can rely on formulas to find the tangent of a sum or a difference of angles. The main difference is that you can't read tangents directly from the coordinates of points on [more…]

How to Combine Reference Angles with Other Techniques to Solve Trigonometric Equations

You can incorporate reference angles into some other pre-calculus techniques to solve trig equations. One such technique is factoring. You've been factoring since algebra, so this process shouldn't be [more…]

How to Graph a Cotangent Function

The parent graphs of tangent and cotangent are comparable because they both have asymptotes and x-intercepts. The only differences you can see are the values of theta where the asymptotes and [more…]

How to Change the Amplitude of a Sine or Cosine Graph

Multiplying a sine or cosine function by a constant changes the graph of the parent function; specifically, you change the amplitude of the graph. When measuring the height of a graph, you measure the [more…]

How to Change the Period of a Sine or Cosine Graph

The period of the parent graphs of sine and cosine is 2 multiplied by pi, which is once around the unit circle. Sometimes in trigonometry, the variable [more…]

How to Shift a Sine or Cosine Graph on the Coordinate Plane

The movement of a parent sine or cosine graph around the coordinate plane is a type of transformation known as a translation or a shift. For this type of transformation, every point on the parent graph [more…]

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