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Trigonometry

Combination of SohCahToa questionsTrigonometry

Find the exact value of trigonometric ratiosTrigonometry

ASTC rule in trigonometry (Trigonometry

Converting between degrees and radiansStill Confused?

Try reviewing these fundamentals first

Trigonometry

Combination of SohCahToa questionsTrigonometry

Find the exact value of trigonometric ratiosTrigonometry

ASTC rule in trigonometry (Trigonometry

Converting between degrees and radiansNope, got it.

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In the world of calculus, pre-calculus, and trigonometry, trigonometric ratios are a powerful and commonly used tool. With proper understanding, they are very easy to use as well!

In order to understand trigonometric ratios, let's first look back on some basic trigonometry principles. First, the Pythagorean Theorem with right triangles, and SOHCAHTOA. If you recall, we can use the Pythagorean Theorem $(a^{2} + b^{2} = c^{2})$ to solve for unknown side lengths of right triangles, and we can use SOHCAHTOA to find missing angles. Below are the formulas we get from SOHCAHTOA, as well as an image to help you visualize it:

Using all of this information, as we further our study into pre-calculus and trigonometry, we will encounter what are known as "trigonometric ratios" – that is, simply just using SOHCAHTOA for important and commonly encountered angles.

Now, let's use the above recalled information to look at what trigonometric ratios are in concrete terms. Below is a table that summarizes the most important trigonometric ratios for $\sin x, \cos x, \tan x, \csc x, \sec x$, and $\cot x$.

This above trig chart gives all the unit circle values for the first quadrant. As you can see, angles are listed in degrees and in radians. You should know both, but you're most likely to be solving problems in radians. Now, the next natural question is, how can I remember these trigonometric ratios? The answer – memorization, and a few helpful little tricks.

Memorizing the common trigonometric ratios in the table above is actually much easier than you would think. Though we can't completely forego memorization, we can make our lives a little easier using this one, incredibly useful trick.

**Trick:**

Because of the following 4 equations, we only need to memorize the unit circle values for sine and cosine.

With these 4 equations, we don't even need to memorize the trig ratios with tan, cot, sec, or csc! Of course, this recommendation will make questions take longer to solve. So, if you're aiming for speed, try memorizing the whole trig chart above!

Now that we have an idea of how to handle trig ratios, the best way to get comfortable with them is to solve some trigonometry problems.

**Example 1:**

Find $\cos 0 + \sin \frac{\pi}{2} + \sec 0$

**Step 1: Evaluate Trigs Individually**

First, let's focus on radians. Let's assume we have only memorized the trigonometric ratios for sin and cos. With this assumption, we know we can easily evaluate the first two trig ratios $\cos 0$ and $\sin \frac{\pi}{2}$. The third ratio, $\sec 0$, however, requires an extra step using the trick we discussed earlier.

**Step 2: Solve**

Now that we've evaluated all of the trig ratios, all we have to do is solve the simple expression above.

**Example 2:**

Find $\sin 90 + \cos 90 - \tan 60$

**Step 1: Evaluate Trigs Individually**

Now let's focus on degrees. Again, let's assume we have only memorized the trigonometric ratios for sin and cos. With this assumption, we know we can again easily evaluate the first two trig ratios easily. The third ratio, however, requires the same trick we used in the previous question, but in this case on $\tan 60$ degrees.

**Step 2: Solve**

Now that we've evaluated all of the trig ratios, all we have to do is solve the simple expression above. Though we try to avoid the use of calculators when using trig ratios, we can certainly use one in this case.

**Example 3:**

Find $\tan \frac{\pi}{3} + \tan \frac{\pi}{2} + \sec \frac{\pi}{4}$

**Step 1: Evaluate Trigs Individually**

For the last example, let's try something a bit more challenging. Again, let's assume we have only memorized the trigonometric ratios for sin and cos. With this assumption, we are still unable to find any of the trig ratios without using our trick, so we have to a bit more work.

**Step 2: Solve**

Now that we've evaluated all of the trig ratios, all we have to do is solve the expression above. However, in this case, it is important to notice that our equation is unsolvable. Since we can't divide by zero, all we need to write in this case is:

And that's it! Hopefully you can now appreciate the simple power of trigonometric ratios. For more practice, visit this link here.

10.

Trigonometry

10.1

Converting between degrees and radians

10.2

Radian measure and arc length

10.3

Angle in standard position

10.4

Coterminal angles

10.5

Reference angle

10.6

Find the exact value of trigonometric ratios

10.7

ASTC rule in trigonometry (All Students Take Calculus)

10.8

Unit circle

10.9

Trigonometric ratios for angles in radians

10.10

Solving first degree trigonometric equations

10.11

Determining non-permissible values for trig expressions

10.12

Use sine ratio to calculate angles and side (Sin = $\frac{o}{h}$ )

10.13

Use cosine ratio to calculate angles and side (Cos = $\frac{a}{h}$ )

10.14

Use tangent ratio to calculate angles and side (Tan = $\frac{o}{a}$ )

10.15

Combination of SohCahToa questions

10.16

Law of sines

10.17

Law of cosines

10.18

Sine graph: y = sin x

10.19

Cosine graph: y = cos x

10.20

Tangent graph: y = tan x

10.21

Cotangent graph: y = cot x

10.22

Secant graph: y = sec x

10.23

Cosecant graph: y = csc x

10.24

Graphing transformations of trigonometric functions

10.25

Determining trigonometric functions given their graphs

10.26

Quotient identities and reciprocal identities

10.27

Pythagorean identities

10.28

Sum and difference identities

10.29

Double-angle identities

10.30

Word problems relating ladder in trigonometry

10.31

Word problems relating guy wire in trigonometry

10.32

Other word problems relating angles in trigonometry

Basic Concepts: Combination of SohCahToa questions, Find the exact value of trigonometric ratios, ASTC rule in trigonometry (**A**ll **S**tudents **T**ake **C**alculus), Converting between degrees and radians

- 1.Find the valuea)sin ${\pi \over 3}$b)cos ${{3 \pi} \over 4}$c)tan ${{7 \pi} \over 6}$d)sin ${{11 \pi} \over 6}$e)sec $-{{8 \pi} \over3}$f)sec $-{{8 \pi} \over3}$ shortcut
- 2.Determine one positive angle and one negative angle that is coterminal with:a)$\theta ={{5 \pi} \over 6}$b)$\theta = - {{8 \pi} \over 3}$
- 3.Use a calculator to determine the value of the trigonometric ratios:a)sin 123°b)sin $123$c)cos $-200$d)tan ${{13 \pi} \over 7 }$e)sin $\pi$f)sin $\pi$°g)sec 500 °

10.

Trigonometry

10.1

Converting between degrees and radians

10.2

Radian measure and arc length

10.3

Angle in standard position

10.4

Coterminal angles

10.5

Reference angle

10.6

Find the exact value of trigonometric ratios

10.7

ASTC rule in trigonometry (All Students Take Calculus)

10.8

Unit circle

10.9

Trigonometric ratios for angles in radians

10.10

Solving first degree trigonometric equations

10.11

Determining non-permissible values for trig expressions

10.12

Use sine ratio to calculate angles and side (Sin = $\frac{o}{h}$ )

10.13

Use cosine ratio to calculate angles and side (Cos = $\frac{a}{h}$ )

10.14

Use tangent ratio to calculate angles and side (Tan = $\frac{o}{a}$ )

10.15

Combination of SohCahToa questions

10.16

Law of sines

10.17

Law of cosines

10.18

Sine graph: y = sin x

10.19

Cosine graph: y = cos x

10.20

Tangent graph: y = tan x

10.21

Cotangent graph: y = cot x

10.22

Secant graph: y = sec x

10.23

Cosecant graph: y = csc x

10.24

Graphing transformations of trigonometric functions

10.25

Determining trigonometric functions given their graphs

10.26

Quotient identities and reciprocal identities

10.27

Pythagorean identities

10.28

Sum and difference identities

10.29

Double-angle identities

10.30

Word problems relating ladder in trigonometry

10.31

Word problems relating guy wire in trigonometry

10.32

Other word problems relating angles in trigonometry

10.1

Converting between degrees and radians

10.2

Radian measure and arc length

10.3

Angle in standard position

10.4

Coterminal angles

10.5

Reference angle

10.6

Find the exact value of trigonometric ratios

10.7

ASTC rule in trigonometry (All Students Take Calculus)

10.9

Trigonometric ratios for angles in radians

10.10

Solving first degree trigonometric equations

10.11

Determining non-permissible values for trig expressions

10.12

Use sine ratio to calculate angles and side (Sin = $\frac{o}{h}$ )

10.13

Use cosine ratio to calculate angles and side (Cos = $\frac{a}{h}$ )

10.14

Use tangent ratio to calculate angles and side (Tan = $\frac{o}{a}$ )

10.15

Combination of SohCahToa questions

10.24

Graphing transformations of trigonometric functions

10.25

Determining trigonometric functions given their graphs

10.26

Quotient identities and reciprocal identities

10.27

Pythagorean identities

10.28

Sum and difference identities

10.29

Double-angle identities

10.30

Word problems relating ladder in trigonometry

10.31

Word problems relating guy wire in trigonometry

10.32

Other word problems relating angles in trigonometry