International Baccalaureate Mathematics

Syllabus Content

Solving trigonometric equations in a finite interval, both graphically and analytically. Equations leading to quadratic equations in sinx, cosx or tanx

Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.

Inverse Trig Calculator This calculator is designed to find all of the angles for a given trigonometric ratio and show them on a graph.

Here are some exam-style questions on this statement:

"(a) Write down the exact value of $\tan 60^o$." ... more

"Solve for x where $ -\pi \le x \le \pi $." ... more

"(a) Show that $2x+15+\dfrac{40}{x-3}= \dfrac{2x^2+9x-5}{x-3}, \quad x \in \mathbb{R}, x \neq 3$" ... more

"(a) Solve the following trigonometric equation for $–360° \lt x \lt 360°$:" ... more

"The cosine of acute angle $ \alpha $ is $ \frac{1}{ \sqrt 5} $" ... more

See all these questions

Here are some Advanced Starters on this statement:

Angle Thinking
Find the range of possible angles, x, for which tan x > cos x > sin x more

Exponential Trigonometry
Find, in degrees, all eight of the solutions to the given exponential, trigonometric equation more

Tansum
Find the sum of the tangents of angles on a straight line. more

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

Trigonometry Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Pupils begin by learning the names on the sides of a right-angled triangle relative to the angles. They then learn the ratios of the lengths of these sides and the connection these ratios have with the size of the angles. Having mastered right-angled triangle trigonometry pupils then progress to more advanced uses including the sine rule and cosine rules. The use of a scientific or graphing calculator is essential for this topic and correct, efficient use of the calculator is an important skill to develop. Here's a Trigonometry Wordsearch just for fun.

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Official Guidance, clarification and syllabus links:

$$2\sin x = 1, \quad 0 \leq x \leq 2\pi$$

Examples:
$$2\sin 2x = 3\cos x, \quad 0^{\circ} \leq x \leq 180^{\circ}$$
$$2\tan(3(x - 4)) = 1, \quad - \pi \leq x \leq 3\pi$$

$$2\sin^2 x + 5\cos x + 1 = 0 \text{ for } 0 \leq x \leq 4\pi,$$
$$2\sin x = \cos 2x, \quad - \pi \leq x \leq \pi$$

Not required: The general solution of trigonometric equations.

Solving trigonometric equations within a finite interval involves finding all the angles that satisfy the equation within a given range. This can be approached graphically, by plotting the functions and identifying the points of intersection, or analytically, by manipulating the equation using trigonometric identities and inverse functions. When trigonometric equations reduce to a quadratic form in $\sin x$, $\cos x$, or $\tan x$, we can solve them using algebraic techniques similar to those used for quadratic equations.

Key Formulae:
$$\sin^2 x + \cos^2 x = 1$$
$$1 + \tan^2 x = \sec^2 x$$
$$\sin(2x) = 2\sin x \cos x$$
$$\cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$$

Example:
To solve the equation $2\sin^2 x - \sin x - 1 = 0$ for $0 \leq x \leq 2\pi$:
Let $u = \sin x$. The equation becomes $2u^2 - u - 1 = 0$.
Solving this quadratic equation, we find $u = -\frac{1}{2}$ or $u = 1$.
Returning to $x$, we have $\sin x = -\frac{1}{2}$ or $\sin x = 1$.
This yields the solutions $x = \frac{7\pi}{6}, \frac{11\pi}{6}$ or $x = \frac{\pi}{2}$ within the given interval.

This video on Solving Trig Functions and Equations is from Revision Village and is aimed at students taking the IB Maths AA SL/HL course.

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International Baccalaureate Mathematics

Geometry and Trigonometry

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