Math Problem Statement

By first finding tan θ, calculate the size of angle θ. Give your answer in degrees to the nearest integer. The triangle has an opposite side of 14 cm and an adjacent side of 9 cm.

Solution

To solve forTo angle solve for \ thetheta angle \ \() \theta in the given right triangle:

Step 1: Identify the sides

We are provided with:

  • Opposite side = 14 cm
  • Adjacent side = 9 cm

We will use the using tangent function the, tangent which ratio (tan⁡θ\tan \theta), we follow these steps:

relates these sides### as Step:

\1[ :\t Recallan the(\ tangenttheta ratio). =\ frac\t{\antext \{thetaop =posite \frac{\text{opposite}}{\text{adjacent}} From the diagram:

  • Opposite side = 14 cm
  • Adjacent side = 9 cm}}

{\Thustext: tan⁡θ=149\tan \theta = \frac{14}{9}

Step 2: Calculate the ratio.

\1] .555--- 6 \, (\text{### Step 2: Calculate \(approximately to 4 decimal places \}).$$ tan--- (\###theta Step 3: Use) the \) \[ \tan(\ inverse tangent ($$\arctan$$) function. theta) = \frac{14}{9}To \ findapprox \( \1theta. \555), we\ calculate] : --- \###[ Step\ theta3 =: \ Findar \(ct \antheta( \1). using555 the6 inverse) tangent\ function] UsingTo a find calculator \( or \ trthetaigon \ometric), table take: the\ **[ ar\ctthetaangent \**approx (\ (\57tan^\^{-circ1 }\\))] of--- ###1 Final. Answer555: : The\ size[ of\ angletheta \( = \ \thetatan \^{-)1 is}( **157.°555**) to\ the] nearestUsing integer a. calculator--- : Do\ you[ want\ furthertheta clarification \ orapprox a breakdown57 of. the5 steps^\?circ Here\ are] --- ### Step 4: Round to the nearest integer fiveR relatedounding questions $$57 you.5$$ to the may nearest integer: $$\theta find = 58^\circ$$ useful: 1. What are the sine and cosine values for the same angle $$\theta$$? 2. How can you calculate the hypotenuse using the Pythagorean theorem? 3. If the triangle were scaled to double--- ### Final Answer: The size of angle $$\theta$$ is **58°**. --- Do you want further clarification or details on the solution? Here are related questions to explore: 1. How do we use sine and cosine to find other angles in a right triangle? 2. What happens if we are given the hypotenuse instead? 3. How can the Pythagorean theorem verify this triangle's measurements? 4. What is the relationship between trigonometric ratios and right angles? 5. How do we convert angles in degrees to radians? **Tip:** Always use the **arctangent function** to find angles when the opposite and adjacent sides are known. its size, how would $$\theta$$ change? 4. What is the relationship between $$\tan \theta$$ and other trigonometric ratios? 5. Can you use trigonometry to solve for $$\theta$$ without knowing any side lengths? ### Tip: For right-angled triangles, the tangent ratio only depends on the opposite and adjacent sides, making it useful when the hypotenuse is unknown.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right-angled triangles
Tangent ratio

Formulas

tan(θ) = opposite / adjacent
θ = arctan(opposite / adjacent)

Theorems

Tangent Function in Right Triangles

Suitable Grade Level

Grades 9-10

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