In these lessons, we will learn

### Special Angles

We will first look into the trigonometric functions of the angles 30˚, 45˚ and 60˚.

**How to find the trig ratios of the special angles?**

Using a 45-45-90 triangle and a 30-60-90 triangle find sine, cosine and tangent values of 0, 30, 45, 60 and 90 degrees**Find exact values of expressions involving sine, cosine and tangent values of 0, 30, 45, 60 and 90 degrees**

Example:

Determine the exact values of each of the following

a) sin30˚tan45˚ + tan30˚sin60˚

b) cos30˚sin45˚ + sin30˚tan30˚

### How to use a calculator to find trig ratios and angles?

We could make use of a scientific calculator to obtain the trigonometric value of an angle. (Your calculator may work in a slightly different way. Please check your manual)

* x* = 6.21 × sin 31.3˚ = 3.2262

**Finding trig ratios and angles using your calculator**

Examples:

1. Use a calculator to find the function value. Use the correct number of significant digits.

a) cos 369.18°

b) tan 426,62°

c) sin 46.6°

d) cot 17.9°

2. Determine θ in degrees. Use the correct number of significant digits.

a) sin θ = 0.42

b) cos θ = 0.29

c) tan θ = 0.91

3. Determine θ in decimal degrees, 0° ≤ θ ≤ 90°. Use the correct number of significant digits.

a) csc θ = 3.6

b) cot θ = 2.1

c) csc θ = 1.63

d) sec θ = 7.25

**Determining Trigonometric Function Values on the Calculator**

Using the TI 84 to find function values for sine, cosine, tangent, cosecant, secant, and cotangent.

Examples:

1. sin 30°

2. cos 45°

3. tan(-264°)

4. sec(102.5°)

5. csc(432°)

6. cot(-23.45°)

**Inverse Trigonometric Functions**
We can use inverse trigonometric functions to find an angle with a given trigonometric value. We can also inverse trigonometric functions to solve a right triangle.

Examples:

1. Use the calculator to find an angle θ in the interval [0, 90] that satisfies the equation.

a) sin θ = 0.7523

b) tan θ = 3.54

2. Solve the given right triangle if a = 44.3 cm and b = 55.9 cm

3. Find eaxh angle in a 3, 4, 5 right triangle.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

- how to find the trigonometric functions of special angles 30˚, 45˚ and 60˚.
- how to use the calculator to evaluate the trigonometric functions of any angle.

Let us consider 30˚ and 60˚.

These two angles form a 30˚-60˚-90˚ right triangle as shown.

The ratio of the sides of the triangle is

From the triangle we get the ratios as follows:

Next, we consider the 45˚ angle that forms a 45˚-45˚-90˚ right triangle as shown. The ratio of the sides of the triangle is

Combining the two tables we get:

* Example: *

Evaluate the following without using a calculator:

a) 2 sin 30˚ + 3 cos 60˚ – 3 tan 45˚

b) 3(cos 30˚)^{2} + 2 (sin 30˚ )^{2}

* Solution: *

a) 2 sin 30˚ + 3 cos 60˚ – 3 tan 45˚

b) 3(cos 30˚)^{2} + 2 (sin 30˚)^{2}

Using a 45-45-90 triangle and a 30-60-90 triangle find sine, cosine and tangent values of 0, 30, 45, 60 and 90 degrees

Example:

Determine the exact values of each of the following

a) sin30˚tan45˚ + tan30˚sin60˚

b) cos30˚sin45˚ + sin30˚tan30˚

* Example: *

Find the value of cos 6.35˚.

* Solution: *

Press

cos 6.35˚ = 0.9939 (correct to 4 decimal places)

*Example: *

Find the value of sin 40˚ 32 ’ .

* Solution: *

sin 40˚32 ’ = 0.6499 (correct to 4 decimal places)

*Example: *

Find the value of x for the following triangle. (Give your answer correct to 4 decimal places)

* Solution: *

Examples:

1. Use a calculator to find the function value. Use the correct number of significant digits.

a) cos 369.18°

b) tan 426,62°

c) sin 46.6°

d) cot 17.9°

2. Determine θ in degrees. Use the correct number of significant digits.

a) sin θ = 0.42

b) cos θ = 0.29

c) tan θ = 0.91

3. Determine θ in decimal degrees, 0° ≤ θ ≤ 90°. Use the correct number of significant digits.

a) csc θ = 3.6

b) cot θ = 2.1

c) csc θ = 1.63

d) sec θ = 7.25

Using the TI 84 to find function values for sine, cosine, tangent, cosecant, secant, and cotangent.

Examples:

1. sin 30°

2. cos 45°

3. tan(-264°)

4. sec(102.5°)

5. csc(432°)

6. cot(-23.45°)

Examples:

1. Use the calculator to find an angle θ in the interval [0, 90] that satisfies the equation.

a) sin θ = 0.7523

b) tan θ = 3.54

2. Solve the given right triangle if a = 44.3 cm and b = 55.9 cm

3. Find eaxh angle in a 3, 4, 5 right triangle.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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