Acute Angles and Right Triangles: Trigonometric Functions and Applications

Acute Angles and Right Triangles

Trigonometric Functions of Acute Angles

Trigonometric functions relate the angles of a right triangle to the ratios of its sides. For any acute angle in a right triangle, the six primary trigonometric functions are defined as follows:

• Sine (sin): Ratio of the side opposite the angle to the hypotenuse.

• Cosine (cos): Ratio of the side adjacent to the angle to the hypotenuse.

• Tangent (tan): Ratio of the side opposite the angle to the side adjacent.

• Cotangent (cot): Reciprocal of tangent.

• Secant (sec): Reciprocal of cosine.

• Cosecant (csc): Reciprocal of sine.

For triangle ABC with sides 77, 36, and 85:

• sin A = $\frac{36}{85}$

• cos A = $\frac{77}{85}$

• tan A = $\frac{36}{77}$

• sin B = $\frac{77}{85}$

• cos B = $\frac{36}{85}$

• tan B = $\frac{77}{36}$

Example: In a right triangle, if the side opposite angle A is 36, the adjacent is 77, and the hypotenuse is 85, then $\sin A = \frac{36}{85}$, $\cos A = \frac{77}{85}$, $\tan A = \frac{36}{77}$.

Special Right Triangles and Trigonometric Values

Special right triangles, such as the 30°-60°-90° triangle, have side ratios that allow for exact trigonometric values:

• For a 30°-60°-90° triangle: sides are 1, $\sqrt{3}$, and 2.

• $\sin 30^\circ = \frac{1}{2}$

• $\cos 30^\circ = \frac{\sqrt{3}}{2}$

• $\tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

• $\cot 30^\circ = \sqrt{3}$

• $\sec 30^\circ = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

• $\csc 30^\circ = 2$

Example: In a 30°-60°-90° triangle, the side opposite 30° is 1, opposite 60° is $\sqrt{3}$, and the hypotenuse is 2.

Trigonometric Functions of Non-Acute Angles

Trigonometric functions can be extended to angles greater than 90°, using reference angles and quadrant analysis. The reference angle is the acute angle formed with the x-axis, and the sign of the function depends on the quadrant.

• Reference angle for 294°: $360^\circ - 294^\circ = 66^\circ$ (Quadrant IV)

• Reference angle for 883°: $883^\circ - 2 \times 360^\circ = 163^\circ$ (Quadrant II)

• Reference angle for 225°: $225^\circ - 180^\circ = 45^\circ$ (Quadrant III)

• Signs in Quadrant III: tan and cot are positive; sin, cos, sec, and csc are negative.

Example: $\sin 225^\circ = -\frac{\sqrt{2}}{2}$, $\cos 225^\circ = -\frac{\sqrt{2}}{2}$, $\tan 225^\circ = 1$.

Finding Trigonometric Function Values Using a Calculator

Calculators are used to find decimal approximations of trigonometric functions for arbitrary angles:

• $\tan 68^\circ 43' \approx 2.567$

• $\cos 193.622^\circ \approx -0.972$

• $\csc 35.8471^\circ \approx 1.708$

• $\sec 287^\circ \approx 3.420$

Example: To find $\cos \theta$ when $\cos \theta \approx 0.921$, solve $\theta \approx 22.9^\circ$.

Solutions and Applications of Right Triangles

Solving Right Triangles

Solving a right triangle involves finding all unknown sides and angles using trigonometric ratios and the Pythagorean theorem.

• Given one side and one acute angle, use sine, cosine, or tangent to find other sides.

• Pythagorean theorem: $a^2 + b^2 = c^2$

Example: In triangle ABC, with angle B = 28°40' and side AB = 25.3 cm:

• $\cos B = \frac{a}{c}$, $a = 25.3$ cm, $c = \frac{25.3}{\cos 28°40'} \approx 28.8$ cm

• $\tan B = \frac{b}{a}$, $b = 25.3 \tan 28°40' \approx 13.8$ cm

Example: In triangle ABC, with sides AC = 44.25 cm and BC = 55.87 cm:

• $c = \sqrt{44.25^2 + 55.87^2} \approx 71.27$ cm

• $\tan A = \frac{44.25}{55.87} \approx 0.792$

• Angle A $\approx 38.38^\circ$

Applications: Height and Distance Problems

Trigonometric functions are used to solve real-world problems involving heights and distances, such as finding the height of a tree or the distance across a river.

Example: To find the height of a tree:

• Given angle B = 60.4°, side adjacent = 15.5 m

• $b = 15.5 \tan 60.4^\circ \approx 27.3$ m

Example: To find the distance using the angle of depression:

• Height = 97 ft, horizontal distance = 42 ft

• $\tan \theta = \frac{97}{42}$, $\theta \approx 66.7^\circ$

Further Applications of Right Triangles

Navigation and Surveying Problems

Trigonometry is essential in navigation and surveying, where distances and angles are used to determine positions and routes.

Example: To find the distance a ship travels:

• Given $\cos 53^\circ = \frac{a}{8.6}$, $a = 8.6 \cos 53^\circ \approx 5.2$ km

Example: In navigation, use the Pythagorean theorem:

• Ship travels 45 nautical miles (b) and 54 nautical miles (a)

• $c = \sqrt{54^2 + 45^2} \approx 70$ nautical miles

Precision Measurement Problems

Trigonometric functions are used in precision measurement, such as determining the distance between two points using small angles.

Example: For small angle $\theta$ and base $b$:

• $d = \frac{b}{2 \cot \theta}$

• Small changes in $\theta$ result in small changes in $d$

Algebraic and Graphical Solutions to Right Triangle Problems

Some right triangle problems require algebraic manipulation or graphical solutions using coordinate axes and slopes.

Example: To find the height of a building:

• Set up equations using tangent for two angles:

• $h = x \tan 74.2^\circ$

• $h = (x + 35) \tan 51.8^\circ$

• Solve for $x$ and $h$ algebraically

Graphical Solution: Use coordinate axes and graph the lines representing the two tangent equations. The intersection gives the height $h$.

Example: The intersection point on the calculator graph gives $x \approx 54.65$ ft and $y \approx 69.45$ ft, which is the height of the building.

Additional info: These notes cover the main concepts and applications from Chapter 2: Acute Angles and Right Triangles, including trigonometric functions, solving triangles, and real-world applications in navigation, surveying, and measurement.