Question

Use the following information. The sum of the angles of a triangle is \(180^{\circ}\). Also, if two angles of a triangle are equal, the lengths of the sides opposite the angles are equal. Depth of a Whale Shark A research ship is tracking the movements of a whale shark that is 700 meters from the ship. The angle formed by the ocean surface and a line from the ship to the whale shark is \(45^{\circ}\). How deep is the whale shark?

Short answer

The whale shark is approximately 495 meters deep.

Step-by-step solution

Identify the right triangle

We first identify the right triangle in the problem. The hypotenuse is the line from the ship to the shark (700 meters). The right angle is at the surface of the ocean and the leg we need to find is the depth of the shark.

Apply the properties of a right triangle

In a right triangle, using the trigonometric function of sine (sin) we know that sin of an angle is equal to the opposite side divided by the hypotenuse. We define the opposite side as the depth of the shark (which we label as 'd') and the hypotenuse as the distance of the shark from the ship (700m). The angle given is \(45^\circ\), hence sine of \(45^\circ\) is equal to d divided by 700 m.

Solve for the unknown 'd'

Rearrange the formula from step 2 to solve for 'd'. This means multiplying both sides of the equation by 700 (the hypotenuse) to isolate 'd'. The formula becomes: d = 700 * sin(45). Sin(45) equals to approximately 0.7071. Therefore, doing the multiplication gives us the depth of the shark.

Calculate the depth of the shark

We can now calculate: d = 700 m * 0.7071. This results in d being approximately equal to 495 meters. Therefore, the whale shark is around 495 metres deep in the ocean.

Key concepts

Right Triangle

When solving trigonometry problems involving a right triangle, it's crucial to identify the triangle's components first. A right triangle has one angle that is exactly 90 degrees, often referred to as the right angle. This special property leads to many practical applications in problem-solving.
In the context of our exercise, the right triangle is formed with the surface of the ocean as one leg and the line from the ship to the shark as the hypotenuse. The key feature of the right triangle is the right angle, which helps distinguish one side as "opposite" the angle of interest, and another side as "adjacent." These sides are central in applying trigonometric functions such as sine, cosine, and tangent.

Sine Function

The sine function is a fundamental tool in trigonometry, used extensively in right triangle calculations. It is defined as the ratio between the length of the side opposite an angle in a right triangle and the length of the hypotenuse. The formula can be expressed as:
\[\sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\]
In our problem, we use the sine of the given angle \(45^{\circ}\) to find "d," the depth of the whale shark. You need to know that \(\sin(45^{\circ})\) is a specific trigonometric value, approximately 0.7071. This value is often memorized or found in trigonometric tables, which aids in calculating sides of right triangles efficiently.

Angle Sum Property

Triangles, including right triangles, have a properties that the sum of their three angles is always \(180^{\circ}\). This is known as the Angle Sum Property. Even though it's not directly used to calculate the depth of the whale shark, it underpins the calculations in trigonometry to ensure that the triangles used are valid for these calculations.
Understanding this property helps in broader contexts where you might need to find missing angles to solve real-world problems. It is quite useful in checking the accuracy of your work or when additional angles are introduced in more advanced problems.

Problem Solving

Problem solving involves breaking down the problem into smaller, manageable parts and systematically solving each part based on known mathematical principles. In our example, identifying the components of the right triangle and selecting the appropriate trigonometric functions are crucial steps.
Here is a simple framework for such problems:

• Identify the right components, like angles, sides, and known values.
• Choose the right trigonometric function for the given angle.
• Rearrange formulas to solve for unknowns.
• Validate answers by checking if they make sense in context.

Ultimately, mastering such problem-solving skills requires practice and familiarity with trigonometric principles, which helps in tackling complex real-world applications effectively.