# Trigonometric Identities Worksheets

Trigonometric identities are equalities, for the most part, that always hold true for all the values of the variables found within it. They are all based on the six trigonometric ratios that are defined using the sides of right triangles. This type of math is used heavily in the architecture of products and buildings. The larger the structure, the more that this math comes in handy. It is used to determine the angles and length of materials that need to be used. Trigonometric identities are critical for properly cost estimating and placed of foundational structures. This collection of worksheets and lessons can be used to learn how to quickly simplify, prove, and solve trigonometric equations and expressions.

### Aligned Standard: HSF-TF.B.5

- From sin to tan values Step-by-step Lesson - See if you can figure out the best way to interchange these values.
- Guided Lesson - You will convert values between: tan and sin, tan and cos, cos and sin.
- Guided Lesson Explanation - At this point, you are expected to use the identities to calculate the outcome.
- Practice Worksheet - There are a lot of conversions between these values.
- Matching Worksheet - Remember to round the values to the nearest hundredths.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

The multiple choice version is what you will commonly see. That is why we highlighted it here.

- Homework 1 - Through every problem we need to remember the Pythagorean Theorem: a
^{2}+ b^{2}= c^{2}. - Homework 2 - We need to remember the trigonometric relationship between tan and sin: tan x = sin x / cos x
- Homework 3 - tan x = a/b, cos x = b/c, sin x = a/c

### Practice Worksheets

These are really order of operations worksheets with just a dash of geometry.

- Practice 1 - If tan x = 9, what is the value of sin x? Rounded to the two decimal places.
- Practice 2 - If tan β =24/12, then what is the value of cos β? Rounded to the nearest hundredth.
- Practice 3 - Use the values to confirm the output.

### Math Skill Quizzes

Some choices can be discredited simply because you can't have the same answer twice.

- Quiz 1 - The reciprocal identities are pretty quick to learn.
- Quiz 2 - The negative angle identities can be used to get rid of negative angles.
- Quiz 3 - The sum of angles identities can help you break apart trig. functions.

### How to Apply Trigonometric Identities

Trigonometric identities are an expression or equation which are true for every variable that is found within it. They are applied to right triangles (those that possess a 90 degree side). If we remember back the hypotenuse is the longest side of a right triangle, the other sides are named based on the angle of interest we are exploring. These identities can be trivially true for like x = x, or at times usefully true for Pythagorean theorems. This type of math is used in many different forms of construction and building especially large structures that need to be well planned prior to construction.

You will find many trigonometric identities, but the common ones are:

tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x), csc(x) = 1/sin(x), sec(x) = 1/cos(x), cot(x) = 1/tan(x), sin(x) = 1/csc(x), cos(x) = 1/sec(x), tan(x) = 1/cot(x), sin2(x) + cos2(x) = 1, cot2A +1 = csc2A, 1 + tan2A = sec2 A

These trigonometric identities are applied in a number of problems. Follow the steps to apply trigonometric identities in a problem.

1. Convert all identities to cos. This can be done using the quotient and reciprocal identities.

2. Look at all the angles to find out the sum and differences and use appropriate identities to remove them. Take your time when you are subtracting and make sure that you have identified the proper minuend and subtrahend.

3. Remove the formulas appropriately and check for angle multiples. If you feel the need; expand the equations, combine the like terms, and then simplify.

4. If you find cos powers greater, replace them with sin powers using the Pythagorean identities.

5. Cancel common factors after factoring numerators and denominators.

6. Now, check if both sides are equal or not.