Lesson Objectives

- Learn how to find trigonometric function values using a calculator
- Learn how to find missing angle measures using inverse trigonometric functions and a calculator

## How to Find Trigonometric Function Values Using a Calculator

In many situations, we will be asked to find trigonometric function values using a calculator. Although this may seem like an easy task, it can be a bit confusing when working with trigonometric functions. Let's start by working with a few values that are already known. We previously learned that the value of cos 60° is 1/2 or 0.5 as a decimal. In order to perform this action, let's first set our calculator to degree mode. Most books recommend using sin 90° as a check. This should yield a value of 1 if you are working with degrees. If you get approximately 0.894, then you are working with radians. Now that we are ready to go, let's hit our cos button followed by 60, this should give you an answer of 0.5 as a decimal or 1/2 as a fraction. Let's look at a few examples.

Example 1: Use a calculator to find each, round to the nearest ten-thousandth.

sin 170°

To find the value here, we hit our sin key followed by 170.

sin 170° ≈ 0.1736

We use the approximately equal to symbol (≈) since we are rounding our answer.

Recall that we can also find this value using the reference angle and the sign rules for trigonometric functions.

Here, our reference angle is 180° - 170° = 10°

Since a 170° angle is in quadrant II sin θ is positive.

sin 170° = sin 10° ≈ 0.1736

Example #2: Use a calculator to find each, round to the nearest ten-thousandth.

sec(-620°)

Most calculators do not have a sec key. Recall from the reciprocal identities lesson, that sec θ = 1/cos θ

To find the value here, we hit 1, followed by the divide key (÷), followed by the cos key, followed by (-620).

sec(-620°) ≈ -5.7588

Again, we could also use the reference angle and our sign rules. The coterminal angle between 0° and 360° is 100°, this means that a -620° angle lies in quadrant II where secant and cosine are negative. The reference angle is 180° - 100°, which is 80°.

sec(-620°) = -sec(80°) ≈ -5.7588

Example #3: Use a calculator to find each, round to the nearest ten-thousandth.

$$\frac{1}{\text{cot}\hspace{.15em}670°}$$ We know from our reciprocal identities that: $$\frac{1}{\text{cot}\hspace{.15em}θ}=\text{tan}\hspace{.15em}θ$$ Since we have a tan key on our calculator, we can hit the tan key followed by 670. $$\frac{1}{\text{cot}\hspace{.15em}670°}≈ -1.1918$$ Again, we could also use the reference angle and our sign rules. The coterminal angle between 0° and 360° is 310°, this means that a 670° angle lies in quadrant IV where tangent and cotangent are negative. The reference angle is 360° - 310°, which is 50°. $$\frac{1}{\text{cot}\hspace{.15em}670°}=-\text{tan}\hspace{.15em}50° ≈ -1.1918$$

Example #4: Use a calculator to find an angle θ in the interval [0°, 90°] that satisfies each condition.

Round to the nearest ten-thousandth. $$\text{sin}\hspace{.1em}θ=\frac{4}{5}$$ To find our unknown angle measure θ, we can use the inverse sine function. $$θ=\text{sin}^{-1}\left(\frac{4}{5}\right) ≈ 53.1301°$$ Example #5: Use a calculator to find an angle θ in the interval [0°, 90°] that satisfies each condition.

Round to the nearest ten-thousandth. $$\text{cos}\hspace{.1em}θ=\frac{2}{7}$$ To find our unknown angle measure θ, we can use the inverse cosine function. $$θ=\text{cos}^{-1}\left(\frac{2}{7}\right) ≈ 73.3985°$$

Example 1: Use a calculator to find each, round to the nearest ten-thousandth.

sin 170°

To find the value here, we hit our sin key followed by 170.

sin 170° ≈ 0.1736

We use the approximately equal to symbol (≈) since we are rounding our answer.

Recall that we can also find this value using the reference angle and the sign rules for trigonometric functions.

Here, our reference angle is 180° - 170° = 10°

Since a 170° angle is in quadrant II sin θ is positive.

sin 170° = sin 10° ≈ 0.1736

Example #2: Use a calculator to find each, round to the nearest ten-thousandth.

sec(-620°)

Most calculators do not have a sec key. Recall from the reciprocal identities lesson, that sec θ = 1/cos θ

To find the value here, we hit 1, followed by the divide key (÷), followed by the cos key, followed by (-620).

sec(-620°) ≈ -5.7588

Again, we could also use the reference angle and our sign rules. The coterminal angle between 0° and 360° is 100°, this means that a -620° angle lies in quadrant II where secant and cosine are negative. The reference angle is 180° - 100°, which is 80°.

sec(-620°) = -sec(80°) ≈ -5.7588

Example #3: Use a calculator to find each, round to the nearest ten-thousandth.

$$\frac{1}{\text{cot}\hspace{.15em}670°}$$ We know from our reciprocal identities that: $$\frac{1}{\text{cot}\hspace{.15em}θ}=\text{tan}\hspace{.15em}θ$$ Since we have a tan key on our calculator, we can hit the tan key followed by 670. $$\frac{1}{\text{cot}\hspace{.15em}670°}≈ -1.1918$$ Again, we could also use the reference angle and our sign rules. The coterminal angle between 0° and 360° is 310°, this means that a 670° angle lies in quadrant IV where tangent and cotangent are negative. The reference angle is 360° - 310°, which is 50°. $$\frac{1}{\text{cot}\hspace{.15em}670°}=-\text{tan}\hspace{.15em}50° ≈ -1.1918$$

## Using Inverse Trigonometric Functions to Find Angles

Previously in our precalculus course, we discussed the concept of the inverse of a function. We know that for a function to have an inverse, it must be a one-to-one function. A trigonometric function like sine is not a one-to-one function but can have an inverse if we restrict the domain. Since we haven't gotten into graphing trigonometric functions yet, we will leave a deep discussion on this topic for later on in the course. For now, we can discuss the basics. We know that a function and its inverse undo each other. We know that our trigonometric functions sine, cosine, and tangent calculate a ratio of two sides based on a given angle measure. When we look at the inverse, we are now calculating the given angle measure based on the ratio of the two sides given. $$\text{sin}\hspace{.1em}θ=\frac{\text{opp}}{\text{hyp}}$$ $$\text{sin}^{-1}\left(\frac{\text{opp}}{\text{hyp}}\right)=θ$$ In other words, with our sine function we plug in an angle measure and get the ratio of opposite side / hypotenuse. With the inverse sine function sin^{-1}, we plug in a ratio of the opposite side / hypotenuse and get the angle measure. We know that sin 30° is 1/2 or 0.5. What would happen if we did sin^{-1}(1/2)? We would get 30° back. $$\text{sin}\hspace{.1em}30°=\frac{1}{2}$$ $$\text{sin}^{-1}\left(\frac{1}{2}\right)=30°$$ Similarly, we have the same result for cosine and tangent: $$\text{cos}\hspace{.1em}θ=\frac{\text{adj}}{\text{hyp}}$$ $$\text{cos}^{-1}\left(\frac{\text{adj}}{\text{hyp}}\right)=θ$$ $$\text{tan}\hspace{.1em}θ=\frac{\text{opp}}{\text{adj}}$$ $$\text{tan}^{-1}\left(\frac{\text{opp}}{\text{adj}}\right)=θ$$ Additionally, we need to know that the inverse of sine is also known as arcsine (arcsin), the inverse of cosine is also known as arccosine (arccos), and the inverse of tangent is known as arctangent (arctan). Let's look at a few examples.Example #4: Use a calculator to find an angle θ in the interval [0°, 90°] that satisfies each condition.

Round to the nearest ten-thousandth. $$\text{sin}\hspace{.1em}θ=\frac{4}{5}$$ To find our unknown angle measure θ, we can use the inverse sine function. $$θ=\text{sin}^{-1}\left(\frac{4}{5}\right) ≈ 53.1301°$$ Example #5: Use a calculator to find an angle θ in the interval [0°, 90°] that satisfies each condition.

Round to the nearest ten-thousandth. $$\text{cos}\hspace{.1em}θ=\frac{2}{7}$$ To find our unknown angle measure θ, we can use the inverse cosine function. $$θ=\text{cos}^{-1}\left(\frac{2}{7}\right) ≈ 73.3985°$$

#### Skills Check:

Example #1

Use a calculator to find each. Round to the nearest ten-thousandth.

cot(-408°)

Please choose the best answer.

A

cot(-408°) ≈ 1.4945

B

cot(-408°) ≈ -0.7431

C

cot(-408°) ≈ -0.9004

D

cot(-408°) ≈ -1.1106

E

cot(-408°) ≈ -1.1108

Example #2

Use a calculator to find an angle θ in the interval [0°, 90°] that satisfies each condition.

Round to the nearest ten-thousandth.

$$\text{cot}\hspace{.2em}θ=\frac{5}{9}$$

Please choose the best answer.

A

θ ≈ 60.9454°

B

θ ≈ 30.2505°

C

θ ≈ 45.0132°

D

θ ≈ 47.92°

E

θ ≈ 51.0352°

Example #3

Use a calculator to find an angle θ in the interval [0°, 90°] that satisfies each condition.

Round to the nearest ten-thousandth.

$$\text{csc}\hspace{.2em}θ=\frac{25}{6}$$

Please choose the best answer.

A

θ ≈ 22.0527°

B

θ ≈ 29.1775°

C

θ ≈ 10.0714°

D

θ ≈ 19.0152°

E

θ ≈ 13.8865°

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