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Tag: tangents

  • Using tangents in Solving Triangles

    Using tangents in Solving Triangles

    if you know the tangent ratio of a given angle, you already know that the ratio is a result of dividing opposite side over tangent. If you know the length of one side, the you can get the other.

    let us say there exits an angle θ that has a tan ratio of 0.821 and the side opposite to it is 8.5 cm. Then the adjacent side can be obtained as follow:

    Example

    Find the length BC in the figure below:

    solution

    The side BC is the opposite side and line AB which is the adjacent side to angle  θ which is equal to  70o is equal to 25cm. Then we proceeds as follows:

    Exercise Questions

    Use tangents to find the lengths of the side marked x in the below tri-angles.

    Q1: Find the unknown side in each of the following

    (a)

    (b)

    Q2: Find the unknown angle θ,α in each of the following triangle

    (a)

    (b)

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  • The tangents of an angle

    The tangents of an angle

    Consider a triangle ABC shown below.

    Angle θ is sitted on line AB and it faces line BC. AB makes a right angle with BC.

    AB is refered as the adjacent side of Angle θ because the θ lies on it.

    Side BC is known as the opposite side of angle θ because θ directly faces it.

    In the figure shown above, BC=8.70cm and Ab=9.90cm.

    I introduce a new line DE, FG, ,IJ, KL and MN all parallel to BC shown.

    The ratio BC/AB = 8.70cm/9.8cm = 0.8876 to 4 decimal places.

    similarly the ratio DE/AE = 7.2cm/8.2cm = 0.8780

    FG/AG = 5.6cm/6.3cm = 0.8888

    IJ/AJ = 4.0cm/4.6cm = 0.8696

    KL/AL = 2.7/3.1 = 0.8710

    MN/AN = 1.7/1.90 = 0.8947

    As can be seen, all the ratios of opposite sides over the adjacent sides is approaching a constant value an all will be 0.9 when rounded to 1 decimal places. Further investigation of ratios of opposite sides versus adjacent sides reveals that such a ratio gives a constant value for a given angle. Such constant value is the tangent of that angle θ and is referred to as the tangent of angle θ usually denoted as tan θ .

    By definition, the tangent of a given angle is the ratio of the opposite side to the adjacent side. That is:

    and in short form:

    Exercise

    Express tan θ as a fraction in the figure below

    Solution

    practice question

    Study the triangle below an express the angle θ as a tan ratio leaving your answer as a fraction.

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    Thank you for your response. ✨

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