Chapter 11 Trigonometry

Practice

11.1 Radian Measure

11.1 Radian Measure Convert the following to radian measures. $\displaystyle 30^{\circ}$ $\displaystyle 45^{\circ}$ $\displaystyle 60^{\circ}$ $\displaystyle 90^{\circ}$ $\displaystyle 120^{\circ}$ $\displaystyle 135^{\circ}$ $\displaystyle 150^{\circ}$ $\displaystyle…

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Practice

11.2 Arc Length

11.2 Arc Length Find the radius $\displaystyle r$ of the circle in the figure. [ In the figure, the arc with length of $\displaystyle…

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Practice

11.3 Area of Sector

11.3 Area of Sector Find the area of each sector. Answer the following. Find the perimeter of a sector of a circle with central…

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Practice

11.4 Advanced Applications

11.4 Advanced Applications For a particular circle with radius $\displaystyle r$, the arc length corresponding to a central angle measuring $\displaystyle x^{\circ}$ is $\displaystyle…

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Practice

11.5 Right-Angled Trigonometry

11.5 Right-Angled Trigonometry $\displaystyle A B C$ is a right triangle in which $\displaystyle C$ is the right angle. If $\displaystyle a=4$ and $\displaystyle…

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11.6 Applications of Trigonometry

11.6 Applications of Trigonometry A ladder is placed along a wall such that its upper end is touching the top of the wall. The…

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11.7 Trigonometric Identities

11.7 Trigonometric Identities Find the exact value of the following: $\displaystyle \cot^3 45^{\circ}+4 \sin^3 30^{\circ}$ $\displaystyle \tan 60^{\circ} \cot 30^{\circ}+4 \sec^2 30^{\circ}$ $\displaystyle \tan^2…

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Practice

11.8 The Laws of Sines and Cosines

11.8 The Laws of Sines and Cosines Calculate the length of side $\displaystyle B C$ in each of the following triangles: Find the size…

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Practice

11.9 Real-World Applications

11.9 Real-World Applications A man standing at a point $\displaystyle P$, sees two trees, $\displaystyle X$ and $\displaystyle Y$, which are respectively $\displaystyle 250…

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Practice

11.10 Area of Oblique Triangles

11.10 Area of Oblique Triangles Calculate the area of the following triangles: What is the area of the triangle $\displaystyle ABC$ if $\displaystyle a…

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Practice

11.11 Complex Circular Problems

11.11 Complex Circular Problems The diagram shows sector $\displaystyle O P Q$ with centre $\displaystyle O$ and sector $\displaystyle P X Y$ with centre…

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Practice

11.12 Trigonometric Ratios of General Angles

11.12 Trigonometric Ratios of General Angles Use the given point on the terminal side of angle $\displaystyle \theta$ to find the value of all…

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11.13 Reduction Formulas

11.13 Reduction Formulas Using the identity $\displaystyle \sin(180^{\circ}-\theta)=\sin \theta$ and related formulas, find the exact values of all trigonometric ratios of: $\displaystyle 120^{\circ}$ $\displaystyle…

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11.14 Trigonometric Equations & Identities

11.14 Trigonometric Equations & Identities Find the exact values of the five trigonometric ratios not given. $\displaystyle \cot \theta=-\sqrt{7},\quad \sin \theta>0$ $\displaystyle \cos \theta=\frac{24}{25},\quad…

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Practice

11.15 Trigonometry in Three Dimensions

11.15 Trigonometry in Three Dimensions Find the length of the longest rod that could be placed in each box shown below. The diagram shows…

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Practice

11.16 Angles in 3D Space

11.16 Angles in 3D Space A telegraph pole is supported by two cables $\displaystyle A D$ and $\displaystyle C D$. The cables are fixed…

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Practice

11.17 Graphs of Trigonometric Functions

11.17 Graphs of Trigonometric Functions The graph below shows the curve with equation $\displaystyle y=\sin 2 x$ in the interval $\displaystyle -60^{\circ} \leq x…

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11.18 Compound Angle Identities

11.18 Compound Angle Identities Using the identities for $\displaystyle \sin(A+B)$ and $\displaystyle \cos(A+B)$, and $\displaystyle \tan A = \frac{\sin A}{\cos A}$: Show that $\displaystyle…

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11.19 Double Angle Identities

11.19 Double Angle Identities Write down an expression for $\displaystyle \sin 2 A$ in terms of $\displaystyle \sin A$ and $\displaystyle \cos A$, and…

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11.20 The Harmonic Form

11.20 The Harmonic Form Express $\displaystyle \cos \left(2 x+45^{\circ}\right)$ in the form $\displaystyle M \cos 2 x+N \sin 2 x$. Solve, for $\displaystyle 0^{\circ}…

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Practice

11.21 Advanced Problem Solving

11.21 Advanced Problem Solving Solving Equations: Solve the following for the given intervals. $\displaystyle 5 \sin \theta-1=0$ for $\displaystyle 0 \leqslant \theta \leqslant \pi$.…

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11.22 Exam Style Problems

11.22 Exam Style Problems Angle \( \displaystyle \alpha \) is acute such that \( \displaystyle \cos \alpha = \frac{3}{5} \). Angle \( \displaystyle \beta…

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